Four distinct quantities are defined by different exponential, radical, or scientific nota… — ACT

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?

1 $5^{20}$
2 $3^{30}$
3 $8.0 \times 10^{14}$
4 $\sqrt{2^{100}}$

Answer

The correct order of the expressions from least to greatest is

5^{20}

, followed by

3^{30}

, then

8.0 \times 10^{14}

, and finally

\sqrt{2^{100}}

Simplifying

\sqrt{2^{100}}

yields

2^{50}

. Expressing

5^{20}

25^{10}

and

3^{30}

27^{10}

shows that

5^{20} < 3^{30}

. By estimating upper and lower bounds relative to powers of

10

, we find that

3^{30} < 30^{10} \approx 5.9 \times 10^{14}

, which is less than

8.0 \times 10^{14}

. Meanwhile,

2^{50} = 1024^5 > (10^3)^5 = 10^{15}

, which is greater than

8.0 \times 10^{14}

. This establishes the sequence

5^{20} < 3^{30} < 8.0 \times 10^{14} < \sqrt{2^{100}}

Step-by-Step Solution

Simplify the radical expression

\sqrt{2^{100}}

using fractional exponent rules.

\sqrt{2^{100}} = (2^{100})^{1/2} = 2^{50}

Converting the square root to a power of

1/2

simplifies the expression into a single base and exponent.

Compare the exponential expressions

5^{20}

and

3^{30}

by expressing them with a common power of

10

5^{20} = (5^2)^{10} = 25^{10}

and

3^{30} = (3^3)^{10} = 27^{10}

. Since

25 < 27

, then

25^{10} < 27^{10}

, so

5^{20} < 3^{30}

Rewriting the powers to share an exponent of

10

allows direct comparison of the base values.

Establish an upper limit for

3^{30}

to compare it with the scientific notation quantity

8.0 \times 10^{14}

3^{30} = 27^{10} < 30^{10} = 3^{10} \times 10^{10}

. Since

3^5 = 243

, we have

3^{10} = 243^2 = 59,049

. Therefore,

3^{10} \times 10^{10} = 5.9049 \times 10^{14}

. Since

5.9049 \times 10^{14} < 8.0 \times 10^{14}

, we have

3^{30} < 8.0 \times 10^{14}

Using a slightly larger round number base of

30

allows an upper bound calculation that demonstrates

3^{30}

is strictly less than

8.0 \times 10^{14}

Establish a lower limit for the simplified radical expression

2^{50}

to compare it with

8.0 \times 10^{14}

2^{50} = (2^{10})^5 = 1024^5

. Since

1024 > 1000 = 10^3

, then

1024^5 > (10^3)^5 = 10^{15}

. Because

10^{15} = 10 \times 10^{14} > 8.0 \times 10^{14}

, it follows that

2^{50} > 8.0 \times 10^{14}

Using the approximation

2^{10} \approx 10^3

shows that

2^{50}

is greater than

10^{15}

, which exceeds the scientific notation value.

Combine the individual inequalities to construct the complete chain of comparisons.

5^{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

5^{20} = 25^{10} = (2.5 \times 10)^{10} = 2.5^{10} \times 10^{10}

. Since

2.5^{10} \approx 9536

, we get

5^{20} \approx 9.5 \times 10^{13}

. Applying a similar approximation to

3^{30} = 27^{10} = 2.7^{10} \times 10^{10} \approx 2.06 \times 10^{14}

. Since

2^{50} = 1.024^5 \times 10^{15} \approx 1.13 \times 10^{15}

, we can compare the coefficients and exponents:

9.5 \times 10^{13} < 2.06 \times 10^{14} < 8.0 \times 10^{14} < 1.13 \times 10^{15}

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic