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Exponents, Roots, and Scientific Notation

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Question 1Question →

Arrange the following mathematical expressions in order from least to greatest value.

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Answer

The correct order of the expressions from least to greatest is 23×522^3 \times 5^2, 4×1024 \times 10^2, 2.5×105\sqrt{2.5 \times 10^5}, and 0.06×1040.06 \times 10^4.
Evaluating each term yields 23×52=2002^3 \times 5^2 = 200, 4×102=4004 \times 10^2 = 400, 2.5×105=500\sqrt{2.5 \times 10^5} = 500, and 0.06×104=6000.06 \times 10^4 = 600. Comparing these simplified values shows that 200<400<500<600200 < 400 < 500 < 600, confirming the correct sequence.

Step-by-Step Solution

1
Evaluate the expression with bases 2 and 5.
23×52=8×25=2002^3 \times 5^2 = 8 \times 25 = 200
First compute the exponential terms and then multiply the resulting values.
2
Evaluate the expression in standard scientific notation.
4×102=4×100=4004 \times 10^2 = 4 \times 100 = 400
Multiply 4 by the value of 10210^2.
3
Evaluate the square root expression.
2.5×105=25×104=5×102=500\sqrt{2.5 \times 10^5} = \sqrt{25 \times 10^4} = 5 \times 10^2 = 500
Adjust the expression under the square root to make it easier to simplify: 2.5×105=25×1042.5 \times 10^5 = 25 \times 10^4. The square root of 25 is 5, and the square root of 10410^4 is 10210^2.
4
Evaluate the non-standard power of ten expression.
0.06×104=0.06×10,000=6000.06 \times 10^4 = 0.06 \times 10,000 = 600
Multiply the decimal coefficient by the power of ten by shifting the decimal point four places to the right.
5
Compare and order all the simplified values.
200<400<500<600200 < 400 < 500 < 600
Sort the values from least to greatest to determine the correct order.

Key Concept

Evaluating expressions involving powers, roots, and scientific notation to compare their values.
Question 2Question →

Which of the following is equivalent to the expression (53)452\frac{(5^3)^4}{5^2}?

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Answer: 5105^{10}

Answer

The correct simplified expression is 5105^{10}.
The expression can be simplified using basic exponent laws. First, using the power of a power rule, the numerator (53)4(5^3)^4 simplifies to 53×4=5125^{3 \times 4} = 5^{12}. Then, using the quotient rule, dividing 5125^{12} by 525^2 simplifies to 512−2=5105^{12 - 2} = 5^{10}. This corresponds to the option containing 5105^{10}.

Step-by-Step Solution

1
Simplify the numerator using the power of a power rule: (am)n=am×n(a^m)^n = a^{m \times n}.
(53)4=53×4=512(5^3)^4 = 5^{3 \times 4} = 5^{12}
When raising a power to another power, multiply the exponents.
2
Simplify the fraction using the quotient rule: aman=am−n\frac{a^m}{a^n} = a^{m - n}.
51252=512−2=510\frac{5^{12}}{5^2} = 5^{12 - 2} = 5^{10}
When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Key Concept

Applying exponent rules (power of a power rule and quotient rule) to simplify exponential expressions.

Alternative Method

Alternatively, you can write out the terms as repeated multiplication: (53)4=53×53×53×53=512(5^3)^4 = 5^3 \times 5^3 \times 5^3 \times 5^3 = 5^{12}, and then divide by 525^2 by cancelling out two factors of 5, which leaves ten factors of 5, or 5105^{10}.
Estimated Time:45s
Question 3Question →

A certain type of single-celled organism has a length of approximately 2.5×10−62.5 \times 10^{-6} meters. If 4×1034 \times 10^3 of these organisms are placed end-to-end in a straight line, what is the total length of the line, in meters, written in scientific notation?

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Answer: 1.0×10−21.0 \times 10^{-2}

Answer

The total length of the line is 1.0×10−21.0 \times 10^{-2} meters.
To find the total length of the line of organisms, multiply the length of a single organism by the total number of organisms: (2.5×10−6)×(4×103)(2.5 \times 10^{-6}) \times (4 \times 10^3). Grouping the coefficients and powers of ten gives (2.5×4)×(10−6×103)=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-3}. To write this in standard scientific notation, rewrite 1010 as 1.0×1011.0 \times 10^1, which yields 1.0×101+(−3)=1.0×10−21.0 \times 10^{1 + (-3)} = 1.0 \times 10^{-2}.

Step-by-Step Solution

1
Set up the multiplication of the two values to find the total length.
Total Length =(2.5×10−6)×(4×103)= (2.5 \times 10^{-6}) \times (4 \times 10^3)
To find the combined length of multiple organisms placed end-to-end, multiply the length of one organism by the total number of organisms.
2
Group the coefficients and the powers of 1010 together, then perform the multiplication.
(2.5×4)×(10−6×103)=10×10−6+3=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-6 + 3} = 10 \times 10^{-3}
Use the associative and commutative properties of multiplication, and add the exponents when multiplying powers with the same base: 10a×10b=10a+b10^a \times 10^b = 10^{a+b}.
3
Convert the result into standard scientific notation a×10na \times 10^n where 1≤∣a∣<101 \le |a| < 10.
10×10−3=1.0×101×10−3=1.0×10−210 \times 10^{-3} = 1.0 \times 10^1 \times 10^{-3} = 1.0 \times 10^{-2}
Since 1010 is not less than 1010, rewrite 1010 as 1.0×1011.0 \times 10^1 and add the exponents (1+(−3)=−21 + (-3) = -2) to express the number in standard scientific notation.

Key Concept

Multiplying numbers in scientific notation and converting to standard scientific notation.
Estimated Time:1m 0s
Question 4Question →

What is the correct order of the following numbers from least to greatest?

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Answer

The correct order from least to greatest is: 6.8×10−56.8 \times 10^{-5}, 7.1×10−47.1 \times 10^{-4}, 1.5×10−31.5 \times 10^{-3}, and 2.4×10−32.4 \times 10^{-3}.
The correct order is determined by first arranging the values by their power of 10 exponent from least to greatest: −5<−4<−3-5 < -4 < -3. This places 6.8×10−56.8 \times 10^{-5} as the smallest and 7.1×10−47.1 \times 10^{-4} as the second smallest. For the remaining two values that share the exponent −3-3, we compare their coefficients: 1.5<2.41.5 < 2.4, placing 1.5×10−31.5 \times 10^{-3} before 2.4×10−32.4 \times 10^{-3}.

Step-by-Step Solution

1
Compare the exponents of the base 10 to establish the primary order of magnitude.
The exponents are −5-5, −4-4, and −3-3. Since −5<−4<−3-5 < -4 < -3, we know that any number with an exponent of −5-5 is smaller than one with −4-4, which in turn is smaller than one with −3-3.
In scientific notation, a smaller exponent on the power of 10 indicates a smaller overall value because it represents shifting the decimal point further to the left.
2
Place the numbers with unique exponents in order.
The smallest number is 6.8×10−56.8 \times 10^{-5}, followed by 7.1×10−47.1 \times 10^{-4}.
These have the smallest exponents (−5-5 and −4-4, respectively).
3
Compare the coefficients of the remaining numbers that share the same exponent.
Both 1.5×10−31.5 \times 10^{-3} and 2.4×10−32.4 \times 10^{-3} share the exponent −3-3. Comparing their coefficients, 1.5<2.41.5 < 2.4, so 1.5×10−3<2.4×10−31.5 \times 10^{-3} < 2.4 \times 10^{-3}.
When the powers of 10 are identical, the values can be compared directly by their coefficients.

Key Concept

Comparing and ordering numbers written in scientific notation by analyzing their powers of 10 and coefficients.
Question 5Question →

A high-speed fiber-optic cable transmits data at a rate of 8.0×1078.0 \times 10^7 bytes per second. A digital archive containing a total of 3.2×10113.2 \times 10^{11} bytes needs to be transmitted. How many seconds will it take to complete the transmission?

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Answer: 4.0×1034.0 \times 10^3

Answer

The transmission will take 4.0×1034.0 \times 10^3 seconds.
The correct answer represents the transmission time of 4.0×1034.0 \times 10^3 seconds. This is determined by dividing the total data volume, 3.2×10113.2 \times 10^{11} bytes, by the rate, 8.0×1078.0 \times 10^7 bytes per second. The division yields 3.28.0×1011−7=0.4×104\frac{3.2}{8.0} \times 10^{11 - 7} = 0.4 \times 10^4. Converting this value to standard scientific notation by moving the decimal point one digit to the right reduces the power of 1010 by one, resulting in 4.0×1034.0 \times 10^3.

Step-by-Step Solution

1
Set up the division to find the transmission time.
Time=3.2×1011 bytes8.0×107 bytes/second\text{Time} = \frac{3.2 \times 10^{11}\text{ bytes}}{8.0 \times 10^7\text{ bytes/second}}
To find the total time, divide the total quantity of data by the transmission rate.
2
Divide the numerical coefficients.
3.28.0=0.4\frac{3.2}{8.0} = 0.4
When dividing numbers written in scientific notation, divide the coefficients first.
3
Apply the quotient rule of exponents to the powers of 10.
1011107=1011−7=104\frac{10^{11}}{10^7} = 10^{11 - 7} = 10^4
Subtract the exponent in the denominator from the exponent in the numerator.
4
Combine the intermediate parts and convert to standard scientific notation.
0.4×104=4.0×1030.4 \times 10^4 = 4.0 \times 10^3
Standard scientific notation requires the coefficient to be at least 1 but strictly less than 10. Shifting the decimal point one place to the right decreases the exponent by 1.

Key Concept

Dividing numbers in scientific notation and converting to standard form
Estimated Time:1m 0s
Question 6Question →

An expression is given as 26⋅34\sqrt{2^6 \cdot 3^4}. What is the value of this expression?

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Answer: 72

Answer

72
The correct answer of 7272 is obtained by applying the rules of exponents under a radical. Evaluating 26⋅34\sqrt{2^6 \cdot 3^4} simplifies to 23⋅322^3 \cdot 3^2, which equals 8⋅9=728 \cdot 9 = 72.

Step-by-Step Solution

1
Express the square root as a fractional exponent of 1/2.
(26⋅34)12(2^6 \cdot 3^4)^{\frac{1}{2}}
The square root of any non-negative number xx can be written as x12x^{\frac{1}{2}}.
2
Apply the power of a product rule (ab)n=anbn(ab)^n = a^n b^n and power of a power rule (am)n=amn(a^m)^n = a^{mn} to simplify the exponents.
26⋅12⋅34⋅12=23⋅322^{6 \cdot \frac{1}{2}} \cdot 3^{4 \cdot \frac{1}{2}} = 2^3 \cdot 3^2
Exponents are multiplied when raising a power to a power.
3
Evaluate the simplified exponential expressions and multiply the results.
8⋅9=728 \cdot 9 = 72
Evaluating 23=82^3 = 8 and 32=93^2 = 9 and calculating their product gives the final simplified value.

Key Concept

Simplifying expressions containing exponents and square roots
Question 7Question →

What is the value of the expression 53−645^3 - \sqrt{64}?

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Answer: 117

Answer

The correct answer is 117117.
Evaluating 535^3 gives 125125 since 5×5×5=1255 \times 5 \times 5 = 125. The square root of 6464 is 88 because 82=648^2 = 64. Subtracting 88 from 125125 yields the correct result of 117117.

Step-by-Step Solution

1
Evaluate the exponential term 535^3
125125
An exponent indicates how many times a base is multiplied by itself. Here, 53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125.
2
Evaluate the radical term 64\sqrt{64}
88
The square root of a number is the non-negative value that, when multiplied by itself, equals the original number. Since 8×8=648 \times 8 = 64, 64=8\sqrt{64} = 8.
3
Subtract the evaluated terms
117117
Subtract the value of the radical from the value of the exponent to simplify the expression: 125−8=117125 - 8 = 117.

Key Concept

Evaluating basic exponents and square roots
Question 8Question →

Arrange the following mathematical values in order from least to greatest.

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Answer

The correct order from least to greatest is 4×10−24 \times 10^{-2}, 2−32^{-3}, 0.09\sqrt{0.09}, and 322×101\frac{3^2}{2 \times 10^1}.
By converting each value to a decimal, we find 4×10−2=0.044 \times 10^{-2} = 0.04, 2−3=0.1252^{-3} = 0.125, 0.09=0.3\sqrt{0.09} = 0.3, and 322×101=0.45\frac{3^2}{2 \times 10^1} = 0.45. Ordering these decimals from smallest to largest yields the correct sequence.

Step-by-Step Solution

1
Convert the scientific notation expression 4×10−24 \times 10^{-2} to its decimal form.
4×10−2=0.044 \times 10^{-2} = 0.04
Multiplying by 10−210^{-2} is equivalent to moving the decimal point of the coefficient 44 two places to the left.
2
Convert the negative exponent expression 2−32^{-3} to a fraction and then to a decimal.
2−3=123=18=0.1252^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125
A negative exponent indicates the reciprocal of the base raised to the positive power.
3
Evaluate the square root of the decimal 0.09\sqrt{0.09}.
0.09=0.3\sqrt{0.09} = 0.3
The square root of a decimal less than 11 results in a larger decimal value because 0.3×0.3=0.090.3 \times 0.3 = 0.09.
4
Evaluate the exponential fraction expression 322×101\frac{3^2}{2 \times 10^1}.
322×101=920=0.45\frac{3^2}{2 \times 10^1} = \frac{9}{20} = 0.45
Simplify the numerator to 99 and the denominator to 2020, then divide to find the decimal equivalent.
5
Compare the evaluated decimal values to determine the correct order from least to greatest.
0.04<0.125<0.3<0.450.04 < 0.125 < 0.3 < 0.45, which correspond to 4×10−2<2−3<0.09<322×1014 \times 10^{-2} < 2^{-3} < \sqrt{0.09} < \frac{3^2}{2 \times 10^1}.
Comparing decimal place values shows that 0.040.04 is the smallest and 0.450.45 is the largest.

Key Concept

Converting and comparing exponential, radical, and scientific notation expressions to a common decimal format.
Question 9Question →

An astronomical unit (AU) is a unit of length equal to approximately 1.50×1081.50 \times 10^8 kilometers. A research probe travels through space at a constant speed of 2.50×1042.50 \times 10^4 meters per second. Which of the following is closest to the number of hours it will take the probe to travel a distance of 5.005.00 AU?

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Answer: 8.33×1038.33 \times 10^3

Answer

The correct answer is 8.33×1038.33 \times 10^3 hours.
To find the travel time in hours, the distance of 5.005.00 AU is first converted to meters: 5.00×1.50×108 km×103 m/km=7.50×1011 m5.00 \times 1.50 \times 10^8\text{ km} \times 10^3\text{ m/km} = 7.50 \times 10^{11}\text{ m}. Dividing this distance by the speed of 2.50×104 m/s2.50 \times 10^4\text{ m/s} yields a time of 3.00×107 seconds3.00 \times 10^7\text{ seconds}. Dividing this time by the 3,600 seconds3,600\text{ seconds} in an hour results in 3.00×1073,600≈8,333.33 hours\frac{3.00 \times 10^7}{3,600} \approx 8,333.33\text{ hours}, which is written in standard scientific notation as 8.33×103 hours8.33 \times 10^3\text{ hours}.

Step-by-Step Solution

1
Convert the total distance from astronomical units (AU) to meters.
Distance = 7.50×10117.50 \times 10^{11} meters
Since 1 AU≈1.50×108 km1\text{ AU} \approx 1.50 \times 10^8\text{ km} and 1 km=103 m1\text{ km} = 10^3\text{ m}, then 1 AU≈1.50×1011 m1\text{ AU} \approx 1.50 \times 10^{11}\text{ m}. Therefore, a distance of 5.00 AU=5.00×(1.50×1011 m)=7.50×1011 m5.00\text{ AU} = 5.00 \times (1.50 \times 10^{11}\text{ m}) = 7.50 \times 10^{11}\text{ m}.
2
Calculate the travel time in seconds.
Time = 3.00×1073.00 \times 10^7 seconds
Dividing the total distance in meters by the speed in meters per second gives: 7.50×1011 m2.50×104 m/s=3.00×107 seconds\frac{7.50 \times 10^{11}\text{ m}}{2.50 \times 10^4\text{ m/s}} = 3.00 \times 10^7\text{ seconds}.
3
Convert the travel time from seconds to hours.
Time ≈8.33×103\approx 8.33 \times 10^3 hours
Since 1 hour=3,600 seconds1\text{ hour} = 3,600\text{ seconds} (which is 3.60×103 seconds3.60 \times 10^3\text{ seconds}), we divide the time in seconds by 3,6003,600: 3.00×1073.60×103≈0.833×104=8.33×103 hours\frac{3.00 \times 10^7}{3.60 \times 10^3} \approx 0.833 \times 10^4 = 8.33 \times 10^3\text{ hours}.

Key Concept

Applying division and unit conversion with numbers written in scientific notation.
Estimated Time:2m 0s
Question 10Question →

What is the value of the expression 23⋅22−812^3 \cdot 2^2 - \sqrt{81}?

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Answer: 23

Answer

23
To evaluate the expression, follow the order of operations. First, simplify the exponential multiplication: 23⋅22=23+2=25=322^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32. Next, evaluate the square root: 81=9\sqrt{81} = 9. Finally, perform the subtraction: 32−9=2332 - 9 = 23. Thus, the correct value is 23.

Step-by-Step Solution

1
Simplify the exponential term 23⋅222^3 \cdot 2^2
25=322^5 = 32
When multiplying exponential terms with the same base, add their exponents: 3+2=53 + 2 = 5.
2
Evaluate the square root term 81\sqrt{81}
99
The square root of 81 is the positive number that, when multiplied by itself, equals 81, which is 9.
3
Perform the subtraction: 32−932 - 9
2323
Subtract 9 from 32 to find the final value of the expression.

Key Concept

Simplifying numerical expressions using laws of exponents, square roots, and order of operations
Question 11Question →

If a=3a = 3 and b=2b = 2, what is the value of the expression a4−b5\sqrt{a^4 - b^5}?

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Answer: 7

Answer

The value of the expression is 7.
Evaluating the exponents yields 34=813^4 = 81 and 25=322^5 = 32. Substituting these values under the radical gives 81−32=49\sqrt{81 - 32} = \sqrt{49}. Evaluating the square root of 49 gives the correct value of 7.

Step-by-Step Solution

1
Evaluate the terms with exponents using the given values a=3a = 3 and b=2b = 2.
a4=34=81a^4 = 3^4 = 81 and b5=25=32b^5 = 2^5 = 32
Exponent operations must be evaluated before performing subtraction or taking the root.
2
Substitute the evaluated terms back into the radical expression and subtract.
81−32=49\sqrt{81 - 32} = \sqrt{49}
Simplifying the expression under the radical is necessary before taking the square root.
3
Find the square root of the simplified value.
49=7\sqrt{49} = 7
Evaluating the square root of 49 completes the simplification of the expression.

Key Concept

Evaluating expressions involving exponents, subtraction, and square roots
Question 12Question →

Evaluate the mathematical expressions below and arrange them in order from least to greatest value.

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Answer

The correct order from least to greatest is 25\sqrt{25}, 232^3, and 1.2×1011.2 \times 10^1.
Evaluating each expression gives 25=5\sqrt{25} = 5, 23=82^3 = 8, and 1.2×101=121.2 \times 10^1 = 12. Comparing these values gives 5<8<125 < 8 < 12, which yields the sequence: 25\sqrt{25}, 232^3, and 1.2×1011.2 \times 10^1.

Step-by-Step Solution

1
Evaluate the square root expression.
25=5\sqrt{25} = 5
Identify the non-negative number that, when multiplied by itself, equals 2525.
2
Evaluate the exponential expression.
23=82^3 = 8
Multiply the base, 22, by itself three times (2×2×22 \times 2 \times 2).
3
Evaluate the expression in scientific notation.
1.2×101=121.2 \times 10^1 = 12
Multiply 1.21.2 by 1010 to shift the decimal point one spot to the right.
4
Order the resulting values from least to greatest.
5<8<125 < 8 < 12, which corresponds to the expressions: 25\sqrt{25}, 232^3, and 1.2×1011.2 \times 10^1.
Compare the numbers 55, 88, and 1212 to establish their relative sizes.

Key Concept

Evaluating basic exponents, square roots, and scientific notation, then ordering their values.
Question 13Question →

What is the value of the expression 144×2−332\frac{\sqrt{144} \times 2^{-3}}{3^2}?

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Answer: 16\frac{1}{6}

Answer

The correct answer is the option representing 16\frac{1}{6}.
The correct answer is obtained by evaluating each part of the expression: the square root 144\sqrt{144} is 1212, the negative exponent 2−32^{-3} is 18\frac{1}{8}, and the denominator 323^2 is 99. Multiplying the numerator terms yields 12×18=1.512 \times \frac{1}{8} = 1.5, and dividing by the denominator 99 gives 1.59=16\frac{1.5}{9} = \frac{1}{6}.

Step-by-Step Solution

1
Simplify the square root term in the numerator, 144\sqrt{144}.
144=12\sqrt{144} = 12
Since 122=14412^2 = 144, the principal square root of 144144 is 1212.
2
Simplify the negative exponent term in the numerator, 2−32^{-3}.
2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}
A negative exponent indicates the reciprocal of the base raised to the positive power, a−n=1ana^{-n} = \frac{1}{a^n}.
3
Calculate the value of the numerator by multiplying the simplified terms.
Numerator =12×18=128=32= 12 \times \frac{1}{8} = \frac{12}{8} = \frac{3}{2}
Multiply the two simplified parts of the numerator.
4
Simplify the denominator, 323^2.
32=93^2 = 9
323^2 means 3×33 \times 3, which equals 99.
5
Divide the simplified numerator by the simplified denominator to find the final value.
3/29=32×9=318=16\frac{3/2}{9} = \frac{3}{2 \times 9} = \frac{3}{18} = \frac{1}{6}
Divide fractions by multiplying by the reciprocal of the denominator.

Key Concept

Evaluating expressions involving square roots, negative integer exponents, and positive integer exponents.
Question 14Question →

Simplify each of the following expressions and arrange them in order of their value from least to greatest.

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Answer

The correct order of the expressions from least to greatest value is: the expression with square root of four times ten to the fourth power (200), the product expression (250), the division expression (300), and finally the cube root expression (400).
Evaluating all four expressions yields the values 200, 400, 300, and 250 respectively. Arranging these values from least to greatest yields 200, 250, 300, and 400.

Step-by-Step Solution

1
Evaluate the first expression: 4.0×104\sqrt{4.0 \times 10^4}.
200200
Using the properties of square roots and exponents, rewrite as 4.0×104=2×102=200\sqrt{4.0} \times \sqrt{10^4} = 2 \times 10^2 = 200.
2
Evaluate the second expression: 6.4×1073\sqrt[3]{6.4 \times 10^7}.
400400
Adjust the scientific notation to make the exponent divisible by 3: 6.4×107=64×1066.4 \times 10^7 = 64 \times 10^6. Then evaluate the cube root: 643×1063=4×102=400\sqrt[3]{64} \times \sqrt[3]{10^6} = 4 \times 10^2 = 400.
3
Evaluate the third expression: 4.8×1051.6×103\frac{4.8 \times 10^5}{1.6 \times 10^3}.
300300
Divide the coefficients 4.81.6=3\frac{4.8}{1.6} = 3 and subtract the exponents in the denominator from the numerator 105−3=10210^{5-3} = 10^2, yielding 3×102=3003 \times 10^2 = 300.
4
Evaluate the fourth expression: (5.0×103)×(5.0×10−2)(5.0 \times 10^3) \times (5.0 \times 10^{-2}).
250250
Multiply the coefficients 5.0×5.0=25.05.0 \times 5.0 = 25.0 and add the exponents 3+(−2)=13 + (-2) = 1. The result is 25.0×101=25025.0 \times 10^1 = 250.
5
Compare the evaluated values to arrange them from least to greatest.
200<250<300<400200 < 250 < 300 < 400
Comparing the values gives the final order: 4.0×104\sqrt{4.0 \times 10^4} (200), followed by (5.0×103)×(5.0×10−2)(5.0 \times 10^3) \times (5.0 \times 10^{-2}) (250), then 4.8×1051.6×103\frac{4.8 \times 10^5}{1.6 \times 10^3} (300), and finally 6.4×1073\sqrt[3]{6.4 \times 10^7} (400).

Key Concept

Applying exponent rules, square and cube root properties, and arithmetic operations on numbers in scientific notation.
Question 15Question →

A certain cloud database stores a total of 3.6×10153.6 \times 10^{15} bytes of data. If this data is distributed equally among 8.0×1048.0 \times 10^4 servers, how many megabytes of data are stored on each server? (Note: 1 megabyte=106 bytes1\text{ megabyte} = 10^6\text{ bytes})

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Answer: 4.5×1044.5 \times 10^4

Answer

4.5×1044.5 \times 10^4 megabytes
The correct answer is the value that correctly represents the data per server in megabytes. Dividing the total data of 3.6×10153.6 \times 10^{15} bytes by the 8.0×1048.0 \times 10^4 servers gives 4.5×10104.5 \times 10^{10} bytes per server. Converting this to megabytes by dividing by 10610^6 gives 4.5×1044.5 \times 10^4 megabytes.

Step-by-Step Solution

1
Calculate the bytes of data stored on each server by dividing the total data by the number of servers.
0.45×1011 bytes=4.5×1010 bytes0.45 \times 10^{11}\text{ bytes} = 4.5 \times 10^{10}\text{ bytes}
Dividing the total bytes (3.6×10153.6 \times 10^{15}) by the total number of servers (8.0×1048.0 \times 10^4) gives the share of data per server. Mathematically, 3.68.0=0.45\frac{3.6}{8.0} = 0.45, and 1015104=1015−4=1011\frac{10^{15}}{10^4} = 10^{15-4} = 10^{11}.
2
Convert the result from bytes to megabytes using the conversion factor 1 megabyte=106 bytes1\text{ megabyte} = 10^6\text{ bytes}.
4.5×104 megabytes4.5 \times 10^4\text{ megabytes}
To convert bytes to megabytes, divide the number of bytes by 10610^6. Since 4.5×1010106=4.5×1010−6=4.5×104\frac{4.5 \times 10^{10}}{10^6} = 4.5 \times 10^{10-6} = 4.5 \times 10^4.

Key Concept

Division of numbers in scientific notation and unit conversion.
Question 16Question →

Let x=2100x = 2^{100}, y=375y = 3^{75}, and z=550z = 5^{50}. Which of the following inequalities correctly represents the relationship among the values of xx, yy, and zz?

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Answer: x<z<yx < z < y

Answer

x<z<yx < z < y
To find the correct relationship, rewrite x=2100x = 2^{100}, y=375y = 3^{75}, and z=550z = 5^{50} with a common exponent. The greatest common divisor of 100100, 7575, and 5050 is 2525. Using the rule (am)n=am⋅n(a^m)^n = a^{m \cdot n}, rewrite the terms: x=(24)25=1625x = (2^4)^{25} = 16^{25}, y=(33)25=2725y = (3^3)^{25} = 27^{25}, and z=(52)25=2525z = (5^2)^{25} = 25^{25}. Comparing the bases shows 16<25<2716 < 25 < 27, which means 1625<2525<272516^{25} < 25^{25} < 27^{25}, so x<z<yx < z < y.

Step-by-Step Solution

1
Find the greatest common divisor (GCD) of the exponents of the three expressions.
The exponents are 100100, 7575, and 5050. The greatest common divisor of these numbers is 2525.
Finding a common exponent allows us to rewrite each expression with the same power so we can compare their bases directly.
2
Rewrite each expression using the power of a power rule: (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
x=2100=(24)25x = 2^{100} = (2^4)^{25}, y=375=(33)25y = 3^{75} = (3^3)^{25}, and z=550=(52)25z = 5^{50} = (5^2)^{25}.
This expresses all three values in the form b25b^{25}, where bb is the base to be evaluated.
3
Evaluate the bases inside the parentheses.
24=162^4 = 16, so x=1625x = 16^{25}; 33=273^3 = 27, so y=2725y = 27^{25}; 52=255^2 = 25, so z=2525z = 25^{25}.
Evaluating the base values simplifies each expression to a single number raised to the power of 2525.
4
Compare the evaluated bases and write the resulting inequality.
Since 16<25<2716 < 25 < 27, it follows that 1625<2525<272516^{25} < 25^{25} < 27^{25}. This corresponds to x<z<yx < z < y.
Since the exponent 2525 is positive, raising larger positive bases to this exponent results in larger values.

Key Concept

Comparing exponential expressions by rewriting them with a common exponent using exponent rules.
Question 17Question →

A supercomputer simulation requires a total of 1.62×10141.62 \times 10^{14} operations. The simulation is divided equally among 12 identical processors running in parallel. If each processor can perform 4.5×1094.5 \times 10^9 operations per second, how many seconds will it take to complete the simulation?

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Answer: 3.0×1033.0 \times 10^3

Answer

3.0×1033.0 \times 10^3
The correct answer is 3.0×1033.0 \times 10^3. To find the total time to complete the simulation, first compute the total operations the 12 parallel processors can perform per second: 12×(4.5×109)=5.4×101012 \times (4.5 \times 10^9) = 5.4 \times 10^{10} operations/second. Next, divide the total number of required operations by this combined rate: 1.62×10145.4×1010=0.3×1014−10=0.3×104=3.0×103\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}} = 0.3 \times 10^{14-10} = 0.3 \times 10^4 = 3.0 \times 10^3 seconds.

Step-by-Step Solution

1
Determine the combined processing rate of the 12 processors working in parallel.
Combined Rate = 12×(4.5×109)=5.4×101012 \times (4.5 \times 10^9) = 5.4 \times 10^{10} operations per second.
Since the 12 processors run in parallel, their individual speeds are added together to find the total rate of operations performed per second.
2
Divide the total number of operations required by the combined rate of all processors to find the time in seconds.
Time = 1.62×10145.4×1010\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}}
Time equals the total workload divided by the combined rate of work.
3
Perform the division and convert the result to standard scientific notation.
Time = 0.3×1014−10=0.3×104=3.0×1030.3 \times 10^{14 - 10} = 0.3 \times 10^4 = 3.0 \times 10^3 seconds.
Subtracting the exponents during division gives 10410^4. The coefficient 0.30.3 is rewritten in standard scientific notation as 3.0×10−13.0 \times 10^{-1}, so 0.3×104=3.0×1030.3 \times 10^4 = 3.0 \times 10^3.

Key Concept

Scientific notation operations and rate division
Question 18Question →

A red blood cell has a diameter of approximately 0.0000080.000008 meters. When this number is written in scientific notation as a×10na \times 10^n, where 1≤a<101 \leq a < 10, what is the value of nn?

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Answer: -6

Answer

The value of the exponent is −6-6.
To write the number 0.0000080.000008 in scientific notation, we shift the decimal point 6 places to the right to get 88. Since the original value is less than 1, the exponent is negative, giving 8×10−68 \times 10^{-6}. The value of nn is therefore −6-6.

Step-by-Step Solution

1
Locate the decimal point and determine how many places it must be shifted to obtain a coefficient between 1 and 10.
The decimal point must be shifted 6 places to the right to get the number 88.
Scientific notation requires the lead coefficient aa to satisfy 1≤a<101 \leq a < 10.
2
Determine the sign and value of the exponent based on the decimal shift.
The exponent nn is −6-6.
Moving the decimal point to the right to write a decimal value less than 1 results in a negative exponent equal to the number of shifts.

Key Concept

Scientific Notation for Decimals Less Than One
Question 19Question →

What is the value of the expression 34⋅3233\frac{3^4 \cdot 3^2}{3^3}?

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Answer: 27

Answer

27
Evaluating the numerator using the product rule of exponents yields 363^6. Then, applying the quotient rule of exponents to divide 363^6 by 333^3 yields 333^3. Evaluating 333^3 gives 2727.

Step-by-Step Solution

1
Simplify the numerator using the product rule of exponents: am⋅an=am+na^m \cdot a^n = a^{m+n}.
34⋅32=34+2=363^4 \cdot 3^2 = 3^{4+2} = 3^6
When multiplying exponential terms with the same base, add their exponents.
2
Simplify the fraction using the quotient rule of exponents: aman=am−n\frac{a^m}{a^n} = a^{m-n}.
3633=36−3=33\frac{3^6}{3^3} = 3^{6-3} = 3^3
When dividing exponential terms with the same base, subtract the exponent in the denominator from the exponent in the numerator.
3
Evaluate the simplified exponential expression.
33=3⋅3⋅3=273^3 = 3 \cdot 3 \cdot 3 = 27
Evaluate the base raised to the power of 3 by multiplying it by itself three times.

Key Concept

Simplifying expressions using the product and quotient rules of exponents
Estimated Time:45s
Question 20Question →

Four distinct quantities are defined by different exponential, radical, or scientific notation expressions. What is the correct order of these quantities from the smallest value to the largest value?

Drag items to arrange them in the correct order

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Answer

The correct order of the expressions from least to greatest is 5205^{20}, followed by 3303^{30}, then 8.0×10148.0 \times 10^{14}, and finally 2100\sqrt{2^{100}}.
Simplifying 2100\sqrt{2^{100}} yields 2502^{50}. Expressing 5205^{20} as 251025^{10} and 3303^{30} as 271027^{10} shows that 520<3305^{20} < 3^{30}. By estimating upper and lower bounds relative to powers of 1010, we find that 330<3010≈5.9×10143^{30} < 30^{10} \approx 5.9 \times 10^{14}, which is less than 8.0×10148.0 \times 10^{14}. Meanwhile, 250=10245>(103)5=10152^{50} = 1024^5 > (10^3)^5 = 10^{15}, which is greater than 8.0×10148.0 \times 10^{14}. This establishes the sequence 520<330<8.0×1014<21005^{20} < 3^{30} < 8.0 \times 10^{14} < \sqrt{2^{100}}.

Step-by-Step Solution

1
Simplify the radical expression 2100\sqrt{2^{100}} using fractional exponent rules.
2100=(2100)1/2=250\sqrt{2^{100}} = (2^{100})^{1/2} = 2^{50}
Converting the square root to a power of 1/21/2 simplifies the expression into a single base and exponent.
2
Compare the exponential expressions 5205^{20} and 3303^{30} by expressing them with a common power of 1010.
520=(52)10=25105^{20} = (5^2)^{10} = 25^{10} and 330=(33)10=27103^{30} = (3^3)^{10} = 27^{10}. Since 25<2725 < 27, then 2510<271025^{10} < 27^{10}, so 520<3305^{20} < 3^{30}.
Rewriting the powers to share an exponent of 1010 allows direct comparison of the base values.
3
Establish an upper limit for 3303^{30} to compare it with the scientific notation quantity 8.0×10148.0 \times 10^{14}.
330=2710<3010=310×10103^{30} = 27^{10} < 30^{10} = 3^{10} \times 10^{10}. Since 35=2433^5 = 243, we have 310=2432=59,0493^{10} = 243^2 = 59,049. Therefore, 310×1010=5.9049×10143^{10} \times 10^{10} = 5.9049 \times 10^{14}. Since 5.9049×1014<8.0×10145.9049 \times 10^{14} < 8.0 \times 10^{14}, we have 330<8.0×10143^{30} < 8.0 \times 10^{14}.
Using a slightly larger round number base of 3030 allows an upper bound calculation that demonstrates 3303^{30} is strictly less than 8.0×10148.0 \times 10^{14}.
4
Establish a lower limit for the simplified radical expression 2502^{50} to compare it with 8.0×10148.0 \times 10^{14}.
250=(210)5=102452^{50} = (2^{10})^5 = 1024^5. Since 1024>1000=1031024 > 1000 = 10^3, then 10245>(103)5=10151024^5 > (10^3)^5 = 10^{15}. Because 1015=10×1014>8.0×101410^{15} = 10 \times 10^{14} > 8.0 \times 10^{14}, it follows that 250>8.0×10142^{50} > 8.0 \times 10^{14}.
Using the approximation 210≈1032^{10} \approx 10^3 shows that 2502^{50} is greater than 101510^{15}, which exceeds the scientific notation value.
5
Combine the individual inequalities to construct the complete chain of comparisons.
520<330<8.0×1014<21005^{20} < 3^{30} < 8.0 \times 10^{14} < \sqrt{2^{100}}
Linking the inequalities using transitive properties determines the correct least-to-greatest order.

Key Concept

Comparing complex numerical expressions containing powers, radicals, and scientific notation by finding common exponents and using base-10 estimation.

Alternative Method

Alternatively, convert all expressions into scientific notation approximations. Note that 520=2510=(2.5×10)10=2.510×10105^{20} = 25^{10} = (2.5 \times 10)^{10} = 2.5^{10} \times 10^{10}. Since 2.510≈95362.5^{10} \approx 9536, we get 520≈9.5×10135^{20} \approx 9.5 \times 10^{13}. Applying a similar approximation to 330=2710=2.710×1010≈2.06×10143^{30} = 27^{10} = 2.7^{10} \times 10^{10} \approx 2.06 \times 10^{14}. Since 250=1.0245×1015≈1.13×10152^{50} = 1.024^5 \times 10^{15} \approx 1.13 \times 10^{15}, we can compare the coefficients and exponents: 9.5×1013<2.06×1014<8.0×1014<1.13×10159.5 \times 10^{13} < 2.06 \times 10^{14} < 8.0 \times 10^{14} < 1.13 \times 10^{15}.
Estimated Time:3m 0s
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