Number and Numeration Alıştırma Soruları — JAMB UTME — Sayfa 5

If $4^{2x + 1} \times 8^{1 - x} = 32^{x - 1}$ , find the value of $x$ .

\frac{5}{2}

\frac{9}{4}

\frac{3}{2}

3

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{2}$

Cevap

The value of

x

is

\frac{5}{2}

.

Converting all terms to base

2

yields

4^{2x+1} = 2^{4x+2}

,

8^{1-x} = 2^{3-3x}

, and

32^{x-1} = 2^{5x-5}

. Applying the law

a^m \times a^n = a^{m+n}

gives

2^{(4x+2)+(3-3x)} = 2^{x+5}

. Equating exponents gives

x+5 = 5x-5

, which solves cleanly to

x = \frac{5}{2}

.

Adım Adım Çözüm

1

Express all terms with a common prime base (base 2)

4 = 2^2

,

8 = 2^3

, and

32 = 2^5

To apply the laws of indices, all terms must share the same base.

2

Apply the power of a power law

(a^m)^n = a^{mn}

to each term

(2^2)^{2x+1} \times (2^3)^{1-x} = (2^5)^{x-1} \implies 2^{2(2x+1)} \times 2^{3(1-x)} = 2^{5(x-1)} \implies 2^{4x+2} \times 2^{3-3x} = 2^{5x-5}

Multiplying the inner exponent by the outer exponent simplifies nested powers.

3

Apply the multiplication law

a^m \times a^n = a^{m+n}

on the left-hand side

2^{(4x+2) + (3-3x)} = 2^{5x-5} \implies 2^{x+5} = 2^{5x-5}

Powers with the same base being multiplied require adding their exponents.

4

Equate the exponents and solve for

x

x + 5 = 5x - 5 \implies 5 + 5 = 5x - x \implies 10 = 4x \implies x = \frac{10}{4} = \frac{5}{2}

Since the bases are equal and non-zero, their exponents must be equal.

Anahtar Kavram

Solving exponential equations by expressing numbers in terms of a common prime base and applying laws of indices.

Soru 82Soru →

In a medical study of $120$ hospital outpatients, $68$ are prescribed Green tea extract ( $G$ ), $52$ are prescribed Herbal infusion ( $H$ ), and $44$ are prescribed Chamomile syrup ( $C$ ). Furthermore, $24$ patients receive both Green tea extract and Herbal infusion, $18$ receive both Green tea extract and Chamomile syrup, $16$ receive both Herbal infusion and Chamomile syrup, while $8$ receive all three prescriptions. How many patients receive at most one of these three prescriptions?

72

78

54

112

Cevabı ve açıklamayı göster

Cevap: $78$

Cevap

The number of patients who receive at most one of the three prescriptions is

78

.

To find the number of patients who receive at most one prescription, we must calculate the number of patients who receive exactly one prescription as well as those who receive none (the complement of the union relative to the universal set of 120 outpatients). The number receiving only Green tea is

34

, only Herbal infusion is

20

, and only Chamomile syrup is

18

, giving

72

patients taking exactly one prescription. The total taking at least one prescription is

114

, leaving

120 - 114 = 6

patients receiving none. Adding these together yields

72 + 6 = 78

.

Adım Adım Çözüm

1

Calculate the number of patients in each exclusive intersection region.

Patients receiving all three

n(G \cap H \cap C) = 8

. Patients receiving only

G \cap H = 24 - 8 = 16

. Patients receiving only

G \cap C = 18 - 8 = 10

. Patients receiving only

H \cap C = 16 - 8 = 8

.

Subtracting the triple intersection isolates the regions representing patients taking exactly two prescriptions.

2

Calculate the number of patients receiving exactly one prescription.

n(G \text{ only}) = 68 - (16 + 10 + 8) = 34

.

n(H \text{ only}) = 52 - (16 + 8 + 8) = 20

.

n(C \text{ only}) = 44 - (10 + 8 + 8) = 18

. Total receiving exactly one

= 34 + 20 + 18 = 72

.

Each set total is reduced by its overlapping intersection regions to yield the single-prescription regions.

3

Determine the number of patients receiving at least one prescription using the inclusion-exclusion principle.

n(G \cup H \cup C) = (68 + 52 + 44) - (24 + 18 + 16) + 8 = 164 - 58 + 8 = 114

.

The principle of inclusion-exclusion accounts for double-counted pairwise intersections and triple-counted intersections.

4

Calculate the number of patients receiving none of the prescriptions.

n((G \cup H \cup C)') = 120 - 114 = 6

.

Subtracting the total taking at least one prescription from the universal set gives the complement count.

5

Sum the patients receiving exactly one prescription and those receiving zero prescriptions.

Total taking at most one

= 72 + 6 = 78

.

'At most one' encompasses both 'zero prescriptions' and 'exactly one prescription'.

Anahtar Kavram

Three-set principle of inclusion-exclusion and complementary sets within a universal set

Soru 83Soru →

Find the value of

x

that satisfies the exponential equation

\frac{4^{x+1} + 4^x + 4^{x-1}}{2^{x+2} + 2^{x+1} + 2^x} = 24

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Factoring

4^{x-1}

from the numerator yields

4^{x-1}(16 + 4 + 1) = 21 \cdot 4^{x-1}

. Factoring

2^x

from the denominator yields

2^x(4 + 2 + 1) = 7 \cdot 2^x

. Dividing the numerical coefficients gives

\frac{21}{7} = 3

. Substituting

4^{x-1} = 2^{2x-2}

into the ratio gives

3 \cdot \frac{2^{2x-2}}{2^x} = 3 \cdot 2^{x-2}

. Setting

3 \cdot 2^{x-2} = 24

leads to

2^{x-2} = 8 = 2^3

, which gives

x - 2 = 3

and therefore

x = 5

.

Adım Adım Çözüm

1

Factor out common terms from the numerator and denominator

Numerator:

4^{x-1}(4^2 + 4^1 + 1) = 21 \cdot 4^{x-1}

. Denominator:

2^x(2^2 + 2^1 + 1) = 7 \cdot 2^x

.

Grouping power terms simplifies sums of exponential expressions.

2

Divide the numerator by the denominator and convert bases

\frac{21 \cdot 4^{x-1}}{7 \cdot 2^x} = 3 \cdot \frac{(2^2)^{x-1}}{2^x} = 3 \cdot \frac{2^{2x-2}}{2^x}

Simplifying

\frac{21}{7} = 3

and expressing base 4 in base 2 allows applying laws of indices.

3

Apply the quotient rule of indices:

\frac{a^m}{a^n} = a^{m-n}

3 \cdot 2^{(2x-2) - x} = 3 \cdot 2^{x-2}

Subtracting exponents of like bases simplifies the fractional index expression.

4

Set the simplified expression equal to 24 and solve for

x

3 \cdot 2^{x-2} = 24 \implies 2^{x-2} = 8 = 2^3 \implies x - 2 = 3 \implies x = 5

Equating the exponents when bases are identical yields the linear equation

x - 2 = 3

.

Anahtar Kavram

Factoring sums of exponential terms and applying the quotient rule of indices

Soru 84Soru →

What is the simplified form of $\frac{6}{3 + \sqrt{3}}$ after rationalizing the denominator?

3 - \sqrt{3}

3 + \sqrt{3}

\sqrt{3}

3 - 2\sqrt{3}

Cevabı ve açıklamayı göster

Cevap: $3 - \sqrt{3}$

Cevap

3 - \sqrt{3}

Multiplying the numerator and denominator by the conjugate

3 - \sqrt{3}

converts the denominator into

3^2 - (\sqrt{3})^2 = 6

. Dividing

6(3 - \sqrt{3})

by

6

yields

3 - \sqrt{3}

.

Adım Adım Çözüm

1

Identify the conjugate of the denominator

The conjugate of

3 + \sqrt{3}

is

3 - \sqrt{3}

.

To eliminate the surd from the denominator, multiply by its conjugate.

2

Multiply the numerator and denominator by the conjugate

\frac{6(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})}

Multiplying by

\frac{3 - \sqrt{3}}{3 - \sqrt{3}}

is equivalent to multiplying by 1.

3

Simplify the denominator using difference of two squares

(3)^2 - (\sqrt{3})^2 = 9 - 3 = 6

The product

(a+b)(a-b) = a^2 - b^2

simplifies the denominator to a rational number.

4

Divide the numerator by the denominator

\frac{6(3 - \sqrt{3})}{6} = 3 - \sqrt{3}

Canceling out the common factor of 6 gives the final simplified surd expression.

Anahtar Kavram

Rationalization of Binomial Denominators

Soru 85Soru →

If $\log_3 x - 4\log_x 3 = 3$ for $x > 1$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 81

Cevap

The value of

x

is 81.

Applying the change of base identity

\log_x 3 = \frac{1}{\log_3 x}

transforms the equation into

\log_3 x - \frac{4}{\log_3 x} = 3

. Substituting

u = \log_3 x

yields

u^2 - 3u - 4 = 0

, which factors as

(u - 4)(u + 1) = 0

. Given

x > 1

,

u

must be positive, giving

u = 4

. Converting back to exponential form gives

x = 3^4 = 81

.

Adım Adım Çözüm

1

Apply the change of base identity

\log_a b = \frac{1}{\log_b a}

to the term

\log_x 3

.

The term becomes

\frac{1}{\log_3 x}

, so the equation is

\log_3 x - \frac{4}{\log_3 x} = 3

.

To express all logarithmic terms in terms of a single common base.

2

Introduce a substitution variable

u = \log_3 x

.

The equation reduces to

u - \frac{4}{u} = 3

.

Simplifies the equation to a manageable algebraic structure.

3

Multiply the entire equation by

u

and rearrange terms.

u^2 - 3u - 4 = 0

Converts the rational expression into standard quadratic form.

4

Factor the quadratic equation.

(u - 4)(u + 1) = 0 \implies u = 4 \text{ or } u = -1

Finds the candidate values for

u

.

5

Filter out invalid roots based on the domain restriction

x > 1

and solve for

x

.

Since

x > 1

,

\log_3 x > 0

, so

u = 4

. Thus

\log_3 x = 4 \implies x = 3^4 = 81

.

Excludes extraneous solutions and evaluates the final exponentiation.

Anahtar Kavram

Logarithmic Change of Base and Quadratic Reduction

Soru 86Soru →

What is the value of $\frac{\log_9 8 \cdot \log_4 25}{\log_3 5}$ ?

\frac{3}{2}

3

\frac{2}{3}

\frac{5}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{2}$

Cevap

\frac{3}{2}

Applying the power identity

\log_{b^k} a^m = \frac{m}{k} \log_b a

yields

\log_9 8 = \frac{3}{2} \log_3 2

and

\log_4 25 = \log_2 5

. Using change of base,

\log_3 2 \cdot \log_2 5 = \log_3 5

. Thus, the numerator simplifies to

\frac{3}{2} \log_3 5

. Dividing by

\log_3 5

leaves

\frac{3}{2}

.

Adım Adım Çözüm

1

Rewrite logarithmic expressions with base powers in terms of prime bases

\log_9 8 = \log_{3^2} (2^3) = \frac{3}{2} \log_3 2

and

\log_4 25 = \log_{2^2} (5^2) = \frac{2}{2} \log_2 5 = \log_2 5

Applying the logarithmic identity

\log_{b^k} (a^m) = \frac{m}{k} \log_b a

simplifies base powers.

2

Multiply the numerator terms together using the change of base rule

(\log_9 8)(\log_4 25) = \left(\frac{3}{2} \log_3 2\right) \cdot (\log_2 5) = \frac{3}{2} (\log_3 2 \cdot \log_2 5) = \frac{3}{2} \log_3 5$

By change of base,

\log_3 2 \cdot \log_2 5 = \frac{\log 2}{\log 3} \cdot \frac{\log 5}{\log 2} = \frac{\log 5}{\log 3} = \log_3 5

.

3

Divide the numerator result by the denominator

\frac{\frac{3}{2} \log_3 5}{\log_3 5} = \frac{3}{2}

The term

\log_3 5

cancels out from numerator and denominator.

Anahtar Kavram

Change of Base and Exponent Rules for Logarithms

Tahmini Süre:1m 30s

Soru 87Soru →

If $\frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}} = a + b\sqrt{15}$ , where $a$ and $b$ are rational numbers, what is the value of $a - b$ ?

-7

7

-4

-2

Cevabı ve açıklamayı göster

Cevap: $-7$

Cevap

-7

Rationalizing

\frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}}

by multiplying top and bottom by

(\sqrt{3} - \sqrt{5})

yields

\frac{11 - 3\sqrt{15}}{-2} = -\frac{11}{2} + \frac{3}{2}\sqrt{15}

. Comparing this with

a + b\sqrt{15}

gives

a = -\frac{11}{2}

and

b = \frac{3}{2}

. Computing

a - b

gives

-\frac{11}{2} - \frac{3}{2} = -7

.

Adım Adım Çözüm

1

Multiply the numerator and denominator by the conjugate of the denominator

(\sqrt{3} - \sqrt{5})

\frac{(2\sqrt{3} - \sqrt{5})(\sqrt{3} - \sqrt{5})}{(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5})}

Rationalizing the denominator requires using the difference of squares.

2

Expand the numerator and simplify the denominator

Numerator:

2(3) - 2\sqrt{15} - \sqrt{15} + 5 = 11 - 3\sqrt{15}

. Denominator:

3 - 5 = -2

.

Multiply terms using FOIL and replace

(\sqrt{3})^2

with

3

and

(\sqrt{5})^2

with

5

.

3

Divide the numerator by the denominator to express in standard form

a + b\sqrt{15}

\frac{11 - 3\sqrt{15}}{-2} = -\frac{11}{2} + \frac{3}{2}\sqrt{15}

Separate the rational term and the surd coefficient.

4

Equate coefficients to find

a

and

b

, then compute

a - b

a = -\frac{11}{2}, b = \frac{3}{2} \implies a - b = -\frac{11}{2} - \frac{3}{2} = -\frac{14}{2} = -7

Calculate the target expression

a - b

using the derived rational values.

Anahtar Kavram

Rationalization of Binomial Denominators and Equating Surd Coefficients

Soru 88Soru →

A geometric progression has a first term of $5$ and a common ratio of $2$ . What is the $6^{\text{th}}$ term of this progression?

160

320

60

15

Cevabı ve açıklamayı göster

Cevap: $160$

Cevap

The

6^{\text{th}}

term of the geometric progression is

160

.

The

n^{\text{th}}

term of a geometric progression is given by

T_n = a r^{n-1}

. Substituting

a = 5

,

r = 2

, and

n = 6

yields

T_6 = 5 \times 2^5 = 5 \times 32 = 160

, which makes

160

the correct value.

Adım Adım Çözüm

1

Identify the given values from the problem statement.

First term

a = 5

, common ratio

r = 2

, and term index

n = 6

.

These parameters are required for the

n^{\text{th}}

term formula of a geometric progression.

2

State the formula for the

n^{\text{th}}

term of a Geometric Progression (GP).

T_n = a r^{n-1}

The

n^{\text{th}}

term of a GP is obtained by multiplying the initial term by the common ratio raised to the power of

n-1

.

3

Substitute the values into the formula and evaluate.

T_6 = 5 \times 2^{6-1} = 5 \times 2^5 = 5 \times 32 = 160

Evaluating

2^5 = 32

and then multiplying by

5

yields the correct

6^{\text{th}}

term.

Anahtar Kavram

Geometric Progression

n^{\text{th}}

Term

Soru 89Soru →

If $\log_x 2 \cdot \log_{\frac{x}{16}} 2 = \log_{\frac{x}{64}} 2$ , what is the sum of all possible real values of $x$ ?

5

12

32

66

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The sum of all possible real values of

x

is 12.

Applying the reciprocal base formula

\log_a b = \frac{1}{\log_b a}

allows us to express the equation as

\frac{1}{\log_2 x} \cdot \frac{1}{\log_2 x - 4} = \frac{1}{\log_2 x - 6}

. Letting

u = \log_2 x

, we obtain

\frac{1}{u(u-4)} = \frac{1}{u-6}

, which simplifies to the quadratic equation

u^2 - 5u + 6 = 0

. Factoring gives

u = 2

or

u = 3

. Solving for

x = 2^u

yields

x = 2^2 = 4

and

x = 2^3 = 8

. The sum of these values is

4 + 8 = 12

.

Adım Adım Çözüm

1

Apply the change of base rule

\log_a b = \frac{1}{\log_b a}

to convert all logarithms to base 2.

\frac{1}{\log_2 x} \cdot \frac{1}{\log_2\left(\frac{x}{16}\right)} = \frac{1}{\log_2\left(\frac{x}{64}\right)}

Converting terms to a common base (base 2) simplifies the variable terms in the denominators.

2

Use the quotient rule of logarithms

\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N

and substitute

u = \log_2 x

.

\log_2\left(\frac{x}{16}\right) = u - 4 \quad \text{and} \quad \log_2\left(\frac{x}{64}\right) = u - 6

Since

\log_2 16 = 4

and

\log_2 64 = 6

, substituting

u = \log_2 x

transforms the expression into an algebraic equation in terms of

u

.

3

Form and solve the algebraic equation for

u

.

\frac{1}{u(u - 4)} = \frac{1}{u - 6} \implies u(u - 4) = u - 6 \implies u^2 - 5u + 6 = 0

Equating the denominators yields a quadratic equation

u^2 - 5u + 6 = 0

.

4

Factor the quadratic equation to find

u

, then solve for

x

.

(u - 2)(u - 3) = 0 \implies u = 2 \text{ or } u = 3 \implies x = 2^2 = 4 \text{ or } x = 2^3 = 8

Converting back from

u = \log_2 x

gives the real roots

x = 4

and

x = 8

.

5

Calculate the sum of the valid solutions.

4 + 8 = 12

The question asks specifically for the sum of all possible real values of

x

.

Anahtar Kavram

Logarithm Change of Base Formula and Logarithmic Equations

Tahmini Süre:2m 0s

Soru 90Soru →

If $24_x \times 13_x = 345_x$ , where $x$ represents a positive integer base, find the value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of the base

x

is 7.

Expanding

24_x

,

13_x

, and

345_x

into base 10 yields

(2x + 4)(x + 3) = 3x^2 + 4x + 5

. Expanding the left side gives

2x^2 + 10x + 12

. Equating and simplifying gives

x^2 - 6x - 7 = 0

, which factors as

(x - 7)(x + 1) = 0

. The positive integer solution greater than 5 is

x = 7

.

Adım Adım Çözüm

1

Convert all base

x

numbers to decimal (base 10) expressions.

24_x = 2x + 4

,

13_x = x + 3

, and

345_x = 3x^2 + 4x + 5

.

Place-value expansion allows algebraic manipulation in standard base 10.

2

Multiply the expanded factors on the left-hand side.

(2x + 4)(x + 3) = 2x^2 + 10x + 12

.

Applying the distributive property of multiplication.

3

Equate the expanded left-hand side to the right-hand side and rearrange into standard quadratic form.

3x^2 + 4x + 5 - (2x^2 + 10x + 12) = 0 \implies x^2 - 6x - 7 = 0

.

Setting the quadratic expression equal to zero enables factoring.

4

Factor the quadratic equation and select the valid base.

(x - 7)(x + 1) = 0 \implies x = 7

(rejecting

x = -1

).

A number base must be a positive integer strictly greater than any individual digit in the given numbers (max digit is 5).

Anahtar Kavram

Solving polynomial equations derived from number base expansion.

Soru 91Soru →

If $\log_2 (x - 1) + \log_4 (x - 1) + \log_{16} (x - 1) = \frac{7}{2}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is

5

.

Converting all terms to base 2 yields

\log_2(x-1) + \frac{1}{2}\log_2(x-1) + \frac{1}{4}\log_2(x-1) = \frac{7}{4}\log_2(x-1)

. Setting

\frac{7}{4}\log_2(x-1) = \frac{7}{2}

gives

\log_2(x-1) = 2

. Exponentiating both sides in base 2 gives

x - 1 = 2^2 = 4

, which results in

x = 5

.

Adım Adım Çözüm

1

Convert each logarithmic term to base 2 using the change of base property.

\log_4(x-1) = \frac{1}{2}\log_2(x-1) \quad \text{and} \quad \log_{16}(x-1) = \frac{1}{4}\log_2(x-1)

Bases 4 and 16 are powers of 2 (

4 = 2^2

and

16 = 2^4

), allowing transformation to a common base.

2

Substitute these equivalent base-2 terms into the original equation.

\log_2(x-1) + \frac{1}{2}\log_2(x-1) + \frac{1}{4}\log_2(x-1) = \frac{7}{2}

This consolidates the equation into a single logarithmic variable,

\log_2(x-1)

.

3

Factor out

\log_2(x-1)

and add the fractional coefficients.

\left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2(x-1) = \frac{7}{4}\log_2(x-1) = \frac{7}{2}

Summing the coefficients

1 + \frac{1}{2} + \frac{1}{4} = \frac{7}{4}

.

4

Isolate

\log_2(x-1)

and solve for

x

.

\log_2(x-1) = 2 \implies x - 1 = 2^2 = 4 \implies x = 5

Multiplying both sides by

\frac{4}{7}

yields

\log_2(x-1) = 2

, and rewriting in exponential form gives

x = 5

.

Anahtar Kavram

Change of base rule for logarithms:

\log_{b^k} a = \frac{1}{k}\log_b a

Soru 92Soru →

What is the simplified form of the surd expression $\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}}$ after rationalizing the denominator?

5 + 2\sqrt{6}

5 + 12\sqrt{6}

5

5\sqrt{6}

Cevabı ve açıklamayı göster

Cevap: $5 + 2\sqrt{6}$

Cevap

5 + 2\sqrt{6}

To rationalize

\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}}

, we multiply both the numerator and denominator by the conjugate of the denominator,

3\sqrt{2} + 2\sqrt{3}

. Expanding the numerator yields

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

. Expanding the denominator gives

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

. Dividing

30 + 12\sqrt{6}

by

6

gives

5 + 2\sqrt{6}

.

Adım Adım Çözüm

1

Identify the conjugate of the denominator

The conjugate of

3\sqrt{2} - 2\sqrt{3}

is

3\sqrt{2} + 2\sqrt{3}

.

Multiplying by the conjugate converts the binomial denominator into a rational number using the difference of squares identity

(a-b)(a+b) = a^2 - b^2

.

2

Multiply the numerator and denominator by the conjugate

\frac{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}{(3\sqrt{2} - 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}

This maintains expression equivalence while eliminating radicals from the denominator.

3

Expand the numerator and denominator independently

Numerator:

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

. Denominator:

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

.

Apply algebraic expansion

(a+b)^2 = a^2 + 2ab + b^2

and simplification of radicals.

4

Divide each term in the numerator by the denominator

\frac{30 + 12\sqrt{6}}{6} = \frac{30}{6} + \frac{12\sqrt{6}}{6} = 5 + 2\sqrt{6}.

Simplify the fraction to express the answer in standard surd form

a + b\sqrt{c}

.

Anahtar Kavram

Rationalization of Binomial Denominators

Soru 93Soru →

Find the value of

x

that satisfies the exponential equation

\sqrt{\frac{125^{x+1}}{5^{x-1}}} = 25^{x-1}

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is 4.

Converting all terms to base 5 yields

5^{x+2}

on the left-hand side and

5^{2x-2}

on the right-hand side. Setting the exponents equal gives

x + 2 = 2x - 2

, which solves to

x = 4

.

Adım Adım Çözüm

1

Express all terms with a common base of 5

125 = 5^3

and

25 = 5^2

Converting terms to prime base 5 allows the application of standard laws of indices.

2

Simplify the fraction inside the square root

\frac{5^{3x+3}}{5^{x-1}} = 5^{(3x+3) - (x-1)} = 5^{2x+4}

Subtract the denominator exponent from the numerator exponent when dividing like bases.

3

Apply the square root as a fractional exponent

\sqrt{5^{2x+4}} = (5^{2x+4})^{1/2} = 5^{x+2}

Taking the square root of a power is equivalent to multiplying the exponent by 1/2.

4

Equate the simplified exponents of both sides

x + 2 = 2x - 2

With identical bases of 5 on both sides, the exponents must be equal.

5

Solve the linear equation for x

x = 4

Rearranging terms gives

2x - x = 2 + 2

, which yields

x = 4

.

Anahtar Kavram

Indices and Laws of Indices

Soru 94Soru →

What is the value of $(17 - 42) \pmod{9}$ expressed in standard non-negative remainder form?

7

2

5

4

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

(17 - 42) \pmod{9}

in standard non-negative remainder form is

2

.

Evaluating

17 - 42

gives

-25

. Reducing

-25

modulo

9

requires finding a non-negative remainder

r

such that

-25 = 9q + r

with

0 \le r < 9

. Choosing

q = -3

yields

-25 = 9(-3) + 2

, making the correct non-negative remainder

2

. Alternatively,

17 \equiv 8 \pmod{9}

and

42 \equiv 6 \pmod{9}

, so

8 - 6 = 2 \pmod{9}

.

Adım Adım Çözüm

1

Perform the subtraction within the parentheses.

17 - 42 = -25

Evaluate the arithmetic expression inside the modulo operation first.

2

Express

-25

in terms of the modulus

9

using the division algorithm

a = qn + r

, where

0 \le r < n

.

-25 = 9 \times (-3) + 2

The remainder

r

must be non-negative (

0 \le r < 9

).

3

Identify the canonical non-negative remainder.

r = 2

Alternatively,

-25 \equiv -7 \pmod{9}

, and adding the modulus gives

-7 + 9 = 2

.

Anahtar Kavram

Modular Arithmetic and Negative Remainder Reduction

Tahmini Süre:45s

Soru 95Soru →

Find the value of

x

that satisfies the exponential equation

\frac{100^{x+1}}{1000^{x-1}} = 10^{2x-1}

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

x

that satisfies the equation is 2.

Rewriting the terms in base 10 gives

\frac{10^{2x+2}}{10^{3x-3}} = 10^{2x-1}

. Using the division rule of indices yields

10^{-x+5} = 10^{2x-1}

. Equating exponents gives

-x+5 = 2x-1

, which simplifies to

3x = 6

, so

x = 2

.

Adım Adım Çözüm

1

Convert each power to base 10

100^{x+1} = (10^2)^{x+1} = 10^{2x+2}

and

1000^{x-1} = (10^3)^{x-1} = 10^{3x-3}

Converting all non-prime composite bases to powers of a common fundamental base allows exponent comparison.

2

Apply the quotient rule of indices to the left side

\frac{10^{2x+2}}{10^{3x-3}} = 10^{(2x+2)-(3x-3)} = 10^{-x+5}

According to the index quotient law

\frac{a^m}{a^n} = a^{m-n}

, subtract the denominator's exponent from the numerator's exponent.

3

Equate exponents of equal bases

-x + 5 = 2x - 1

If

a^f(x) = a^g(x)

for

a > 0, a \neq 1

, then

f(x) = g(x)

.

4

Solve the linear equation for

x

3x = 6 \implies x = 2

Isolate the variable

x

to find its value.

Anahtar Kavram

Exponential equations solvable by converting to a common base

Soru 96Soru →

Calculate the numerical values of the following sequence and progression quantities, then arrange them in ascending order (from smallest to largest):

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order of the numerical values is: the common ratio of the GP (value = 3), the common difference of the AP (value = 4), the 5th term of the AP (value = 8), and the sum to infinity of the GP (value = 10).

Evaluating each sequence property yields numerical values: the GP common ratio equals 3, the AP common difference equals 4, the AP 5th term equals 8, and the GP sum to infinity equals 10. Ordering these values from least to greatest produces the sequence 3, 4, 8, 10.

Adım Adım Çözüm

1

Calculate the value for the first quantity (AP common difference)

d = 4

From

a + 6d = 27

and

a + 2d = 11

, subtract to find

4d = 16 \implies d = 4

.

2

Calculate the value for the second quantity (GP common ratio)

r = 3

From

ar^4 = 162

and

ar = 6

, divide to obtain

r^3 = 27 \implies r = 3

.

3

Calculate the value for the third quantity (GP sum to infinity)

S_{\infty} = 10

Apply the sum to infinity formula

S_{\infty} = \frac{a}{1-r} = \frac{5}{1 - 0.5} = 10

.

4

Calculate the value for the fourth quantity (AP 5th term)

T_5 = 8

Apply the nth term formula

T_5 = a + 4d = 2 + 4(1.5) = 8

.

5

Sort the calculated values in ascending order

3 < 4 < 8 < 10

Comparing the values gives

3

(GP common ratio),

4

(AP common difference),

8

(AP 5th term), and

10

(GP sum to infinity).

Anahtar Kavram

Nth term, common difference, common ratio, and sum to infinity calculations for AP and GP sequences.

Soru 97Soru →

A student was asked to approximate the fraction $\frac{5}{16}$ as a decimal rounded to $2$ significant figures. What is the percentage error introduced by this approximation?

$0.8\%

$0.81\%

$0.16\%

$0.008\%

Cevabı ve açıklamayı göster

Cevap: $0.8\%

Cevap

The percentage error introduced by the approximation is

0.8\%

.

Converting

\frac{5}{16}

gives

0.3125

. Rounding to

2

significant figures gives an estimated value of

0.31

. The absolute error is

|0.31 - 0.3125| = 0.0025

. Dividing by the true value

0.3125

yields a relative error of

0.008

, which equals

0.8\%

when multiplied by

100\%

.

Adım Adım Çözüm

1

Convert the fraction to an exact decimal

\frac{5}{16} = 0.3125

To find the true value of the number before approximation.

2

Round the exact decimal to 2 significant figures

0.3125 \approx 0.31

The first non-zero digits are 3 and 1; the next digit (2) is less than 5, so we round down.

3

Calculate the absolute error

|0.31 - 0.3125| = 0.0025

Absolute error is the magnitude of the difference between actual and estimated values.

4

Calculate the percentage error

\frac{0.0025}{0.3125} \times 100\% = 0.8\%

Percentage error is given by

\frac{\text{Absolute Error}}{\text{True Value}} \times 100\%

.

Anahtar Kavram

Fractions, Decimals, Percentages, and Approximations

Tahmini Süre:1m 15s

Soru 98Soru →

What is the simplified form of the expression $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} - \sqrt{3}$ ?

2

2 + 2\sqrt{3}

\sqrt{3} - 1

2\sqrt{3}

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The simplified form of the expression is

2

.

Multiplying the numerator and denominator of

\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}

by the conjugate

(\sqrt{6} + \sqrt{2})

gives

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

. Subtracting

\sqrt{3}

yields

2

.

Adım Adım Çözüm

1

Rationalize the denominator of

\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}

Multiply numerator and denominator by the conjugate

(\sqrt{6} + \sqrt{2})

to get

\frac{(\sqrt{6} + \sqrt{2})^2}{(\sqrt{6})^2 - (\sqrt{2})^2}

.

Rationalizing eliminates the surd from the denominator using the difference of two squares identity.

2

Expand the numerator and simplify the fraction

Numerator:

(\sqrt{6})^2 + 2\sqrt{12} + (\sqrt{2})^2 = 6 + 4\sqrt{3} + 2 = 8 + 4\sqrt{3}

. Denominator:

6 - 2 = 4

. Fraction simplifies to

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

.

Simplifying radical factors (

\sqrt{12} = 2\sqrt{3}

) allows division by the common denominator.

3

Subtract

\sqrt{3}

from the rationalized term

(2 + \sqrt{3}) - \sqrt{3} = 2

.

Subtracting like surd terms cancels out

\sqrt{3}

leaving the rational constant.

Anahtar Kavram

Rationalization of Binomial Denominators and Surd Simplification

Soru 99Soru →

What is the value of $x$ that satisfies the equation $16^{x - 1} \times 4^{x + 2} = 8^{x + 3}$ ?

3

\frac{1}{3}

\frac{8}{3}

\frac{3}{2}

Cevabı ve açıklamayı göster

Cevap: $3$

Cevap

The value of

x

is

3

.

Converting all terms to base 2 gives

2^{4(x-1)} \times 2^{2(x+2)} = 2^{3(x+3)}

. Expanding the exponents gives

2^{4x-4} \times 2^{2x+4} = 2^{3x+9}

, which simplifies to

2^{6x} = 2^{3x+9}

. Equating exponents yields

6x = 3x + 9

, giving the solution

x = 3

.

Adım Adım Çözüm

1

Express all terms with a common base of 2.

16 = 2^4

,

4 = 2^2

, and

8 = 2^3

, so the equation becomes

(2^4)^{x-1} \times (2^2)^{x+2} = (2^3)^{x+3}

.

Converting all terms to a common prime base allows application of index laws.

2

Apply power of a power law

(a^m)^n = a^{m n}

.

2^{4x - 4} \times 2^{2x + 4} = 2^{3x + 9}

.

Multiply exponents when raising a power to another power.

3

Apply multiplication law

a^m \times a^n = a^{m+n}

on the left side.

2^{(4x - 4) + (2x + 4)} = 2^{6x}

.

Add exponents when multiplying powers with the same base.

4

Equate exponents of equal bases.

6x = 3x + 9 \implies 3x = 9 \implies x = 3

.

If

a^m = a^n

for

a > 0

and

a \neq 1

, then

m = n

.

Anahtar Kavram

Laws of Indices and Solving Exponential Equations

Soru 100Soru →

If $132_k + 43_k = 230_k$ , where $k$ represents a positive integer base, what is the value of $k$ ?

4

5

6

7

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of the base

k

is 5.

Expanding

132_k

,

43_k

, and

230_k

into base 10 yields

(k^2 + 3k + 2) + (4k + 3) = 2k^2 + 3k

. Simplifying this gives the quadratic equation

k^2 - 4k - 5 = 0

, which factors as

(k - 5)(k + 1) = 0

. Since a number base must be a positive integer greater than the maximum digit present (which is 4), the correct base is 5.

Adım Adım Çözüm

1

Expand each term in the equation

132_k + 43_k = 230_k

into base 10 using place-value notation.

132_k = 1 \cdot k^2 + 3 \cdot k^1 + 2 \cdot k^0 = k^2 + 3k + 2

43_k = 4 \cdot k^1 + 3 \cdot k^0 = 4k + 3

230_k = 2 \cdot k^2 + 3 \cdot k^1 + 0 \cdot k^0 = 2k^2 + 3k

Converting all terms to base 10 allows standard algebraic operations to be performed.

2

Set up the algebraic equation in base 10 and combine like terms.

(k^2 + 3k + 2) + (4k + 3) = 2k^2 + 3k

k^2 + 7k + 5 = 2k^2 + 3k

Equate the sum of the expanded left-hand side terms to the expanded right-hand side.

3

Rearrange the terms to form a standard quadratic equation

ak^2 + bk + c = 0

.

2k^2 + 3k - (k^2 + 7k + 5) = 0

k^2 - 4k - 5 = 0

Subtract

k^2 + 7k + 5

from both sides to set the quadratic expression equal to zero.

4

Factor the quadratic equation to find the possible values of

k

.

(k - 5)(k + 1) = 0 \implies k = 5 \text{ or } k = -1

Solve for

k

using factorization.

5

Select the valid base based on the domain of number bases.

k = 5

(since a number base must be a positive integer greater than the largest digit 4 present in the expression).

Base numbers cannot be negative or smaller than any of their constituent digits.

Anahtar Kavram

Number Bases and Conversions

Tahmini Süre:1m 30s

Number and Numeration

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Anahtar Kavram