Surds and Indices Practice Questions — State PSC Exam

Question 1Question →

If $x = \sqrt{7 + 4\sqrt{3}}$ , $y = \sqrt{7 - 4\sqrt{3}}$ , and $z = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$ , what is the value of the expression $\frac{x^3 + y^3}{x^2 + y^2 - z}$ ?

4

\frac{56}{13}

\frac{13}{3}

13

Show answer & explanation

Answer: 4

Answer

4

Expressing

7 \pm 4\sqrt{3}

as perfect squares

(2 \pm \sqrt{3})^2

simplifies

x

to

2 + \sqrt{3}

and

y

to

2 - \sqrt{3}

. This yields

x+y=4

,

xy=1

,

x^2+y^2=14

, and

x^3+y^3=52

. For

z

, using the cubic identity

z^3 = a^3 + b^3 + 3ab z

transforms the expression into

z^3 + 3z - 4 = 0

, yielding the real solution

z = 1

. Substituting these values gives

\frac{52}{14 - 1} = 4

.

Step-by-Step Solution

1

Simplify the square root surds

x

and

y

x = 2 + \sqrt{3}

and

y = 2 - \sqrt{3}

Since

7 + 4\sqrt{3} = 4 + 3 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2

, taking the square root gives

2 + \sqrt{3}

. Similarly,

7 - 4\sqrt{3} = (2 - \sqrt{3})^2

.

2

Calculate fundamental algebraic combinations of

x

and

y

x + y = 4

,

xy = 1

,

x^2 + y^2 = 14

, and

x^3 + y^3 = 52

x + y = (2+\sqrt{3}) + (2-\sqrt{3}) = 4

.

xy = (2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1

.

x^2 + y^2 = (x+y)^2 - 2xy = 16 - 2 = 14

.

x^3 + y^3 = (x+y)^3 - 3xy(x+y) = 64 - 3(1)(4) = 52

.

3

Evaluate the nested cube root expression for

z

z = 1

Let

z = a + b

where

a = \sqrt[3]{2+\sqrt{5}}

and

b = \sqrt[3]{2-\sqrt{5}}

. Cubing both sides:

z^3 = a^3 + b^3 + 3ab(a+b) = (2+\sqrt{5}) + (2-\sqrt{5}) + 3\sqrt[3]{(2+\sqrt{5})(2-\sqrt{5})} z = 4 + 3\sqrt[3]{4-5} z = 4 - 3z

. Solving

z^3 + 3z - 4 = 0

gives

(z-1)(z^2 + z + 4) = 0

, whose unique real root is

z = 1

.

4

Substitute all values into the given expression

\frac{x^3 + y^3}{x^2 + y^2 - z}

\frac{52}{14 - 1} = \frac{52}{13} = 4

Direct numerical evaluation of the simplified algebraic components.

Key Concept

Simplification of Nested Square and Cube Surds using Algebraic Identities

Estimated Time:3m 0s

Question 2Question →

If $x = \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}$ and $y = \frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}$ , what is the value of $x^2 + y^2$ ?

Show answer & explanation

Answer: 23

Answer

The value of

x^2 + y^2

is 23.

By rationalizing the denominators of

x

and

y

, we obtain

x = \frac{5+\sqrt{21}}{2}

and

y = \frac{5-\sqrt{21}}{2}

. Adding these gives

x+y=5

, and multiplying them gives

xy=1

. Using the identity

x^2+y^2 = (x+y)^2 - 2xy

, we calculate

5^2 - 2(1) = 23

.

Step-by-Step Solution

1

Rationalize the expressions for

x

and

y

by multiplying the numerator and denominator by their respective conjugates.

x = \frac{(\sqrt{7}+\sqrt{3})^2}{7-3} = \frac{10 + 2\sqrt{21}}{4} = \frac{5 + \sqrt{21}}{2}

and

y = \frac{(\sqrt{7}-\sqrt{3})^2}{7-3} = \frac{10 - 2\sqrt{21}}{4} = \frac{5 - \sqrt{21}}{2}

Eliminating surds from the denominator simplifies addition and multiplication.

2

Calculate the sum

(x + y)

and the product

(x \cdot y)

of

x

and

y

.

x + y = \frac{5 + \sqrt{21}}{2} + \frac{5 - \sqrt{21}}{2} = 5

and

x \cdot y = \frac{(5 + \sqrt{21})(5 - \sqrt{21})}{4} = \frac{25 - 21}{4} = 1

Simplifying the symmetric expressions reduces computation complexity.

3

Apply the algebraic identity

x^2 + y^2 = (x + y)^2 - 2xy

.

x^2 + y^2 = (5)^2 - 2(1) = 25 - 2 = 23

Using the identity avoids evaluating complex squares directly.

Key Concept

Rationalization of surds and application of symmetric algebraic identities

Question 3Question →

If $2^{x-1} + 2^{x+1} = 320$ , what is the value of $x$ ?

5

6

7

8

Show answer & explanation

Answer: $7$

Answer

7

Factoring

2^{x-1}

from the left side gives

2^{x-1}(1 + 2^2) = 5 \cdot 2^{x-1} = 320

. Dividing both sides by

5

yields

2^{x-1} = 64 = 2^6

. Equating the powers gives

x - 1 = 6

, which solves to

x = 7

.

Step-by-Step Solution

1

Factor out the common exponential term from the left-hand side

2^{x-1}(1 + 2^2) = 320 \implies 2^{x-1}(1 + 4) = 320 \implies 5 \cdot 2^{x-1} = 320

Applying the distributive law of exponents to simplify the sum.

2

Divide both sides by 5

2^{x-1} = \frac{320}{5} = 64

Isolate the exponential term with base 2.

3

Express 64 as a power of 2 and equate the exponents

64 = 2^6 \implies x - 1 = 6 \implies x = 7

Since the bases are equal (

2

), the exponents must be equal.

Key Concept

Laws of Indices and Exponential Equations

Question 4Question →

Consider the exponential equation:

4^x - 3 \cdot 2^{x+2} + 32 = 0

Determine the sum of all real values of $x$ that satisfy this equation.

Show answer & explanation

Answer: 5

Answer

5

By applying the laws of indices, the original expression transforms into a quadratic equation in terms of

2^x

. Solving

y^2 - 12y + 32 = 0

yields

y=4

and

y=8

, which correspond precisely to

x=2

and

x=3

. Their sum is 5.

Step-by-Step Solution

1

Express all terms with a common base of 2.

4^x

becomes

(2^x)^2

and

2^{x+2}

becomes

4 \cdot 2^x

.

Creating a common base allows the equation to be transformed into a standard polynomial form.

2

Rewrite the equation using the new terms.

(2^x)^2 - 12(2^x) + 32 = 0

Simplifying the coefficients makes it easier to spot the quadratic structure.

3

Perform a substitution to solve the quadratic equation.

Letting

y = 2^x

gives

y^2 - 12y + 32 = 0

. Factoring yields

(y - 4)(y - 8) = 0

, so

y = 4

or

y = 8

.

Substitution converts a complex exponential equation into a simple quadratic one.

4

Solve for the original variable

x

.

2^x = 4 \implies x = 2

, and

2^x = 8 \implies x = 3

.

The question asks for the values of

x

, not the intermediate substitution variable

y

.

5

Calculate the sum of all valid

x

values.

2 + 3 = 5

This addresses the specific final requirement of the question stem.

Key Concept

Solving exponential equations reducible to quadratics using index laws.

Estimated Time:1m 30s

Question 5Question →

Consider the algebraic fraction below:

\frac{5^{n+2} - 5^n}{5^{n+1} - 5^n}

Which of the following represents the simplified exact value of this expression?

4

5

6

25

Show answer & explanation

Answer: 6

Answer

6

The expression can be systematically simplified by factoring out the lowest common power of the base, which is

5^n

, from both the numerator and the denominator. The numerator becomes

5^n(5^2 - 1) = 5^n(24)

. The denominator becomes

5^n(5^1 - 1) = 5^n(4)

. Canceling the common factor

5^n

from the top and bottom leaves

24 / 4

, which equals 6.

Step-by-Step Solution

1

Identify the greatest common factor in both the numerator and the denominator.

The common factor is

5^n

.

Factoring out the lowest power of the base simplifies algebraic expressions involving exponential terms.

2

Factor out

5^n

from the numerator.

5^{n+2} - 5^n = 5^n(5^2 - 1)

Applying the exponent rule

a^{x+y} = a^x \times a^y

in reverse allows us to extract the common term.

3

Factor out

5^n

from the denominator.

5^{n+1} - 5^n = 5^n(5^1 - 1)

Applying the same extraction process to the bottom expression.

4

Substitute the factored forms back into the fraction and cancel the common

5^n

term.

\frac{5^n(25 - 1)}{5^n(5 - 1)} = \frac{24}{4}

The

5^n

multiplier in the numerator and denominator cancel each other out, leaving basic arithmetic values.

5

Perform the final arithmetic division.

24 \div 4 = 6

Simplifying the remaining numerical fraction yields the final exact value.

Key Concept

Simplifying exponential expressions by factoring out common base powers.

Estimated Time:1m 15s

Question 6Question →

If the expression $\frac{3\sqrt{2} - 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}}$ can be expressed in the form $a - b\sqrt{6}$ where $a$ and $b$ are rational numbers, what is the exact value of $a + b$ ?

Show answer & explanation

Answer: 7

Answer

The correct value is 7.

The correct answer is derived by multiplying the numerator and denominator by the conjugate

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

. This rationalizes the denominator to

6

. Expanding the numerator gives

30 - 12\sqrt{6}

. Dividing the numerator by

6

yields

5 - 2\sqrt{6}

. Setting this equal to

a - b\sqrt{6}

identifies

a = 5

and

b = 2

, giving a final sum of

7

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator,

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

.

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

This process, known as rationalizing the denominator, removes the surds from the bottom of the fraction.

2

Expand the numerator using the binomial square formula

(x-y)^2 = x^2 - 2xy + y^2

.

(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 squared binomial simplifies the top part of the fraction.

3

Expand the denominator using the difference of squares formula

(x+y)(x-y) = x^2 - y^2

.

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

This guarantees that the denominator becomes a rational number.

4

Divide the terms in the numerator by the denominator.

\frac{30 - 12\sqrt{6}}{6} = 5 - 2\sqrt{6}

Simplifying the fraction allows us to match it to the given form

a - b\sqrt{6}

.

5

Equate the simplified expression to

a - b\sqrt{6}

and solve for

a + b

.

a = 5

,

b = 2

, and

a + b = 7

By direct comparison of rational and irrational parts, we determine the values of

a

and

b

to find their sum.

Key Concept

Rationalizing the denominator using conjugates and expanding binomial expressions involving surds.

Question 7Question →

If $M = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}}$ and $x$ satisfies the exponential equation $\left(\sqrt[3]{4}\right)^{2x-1} = \left(\frac{1}{16}\right)^{x-4}$ with $\left(\sqrt[3]{4}\right)^{2x-1} = 2^k$ , what is the value of $M^2 + k$ ?

Show answer & explanation

Answer: 23.5

Answer

The value of

M^2 + k

is 23.5.

By writing

7 \pm 2\sqrt{10}

as

(\sqrt{5} \pm \sqrt{2})^2

, the radical simplifies cleanly to

M = 2\sqrt{5}

, giving

M^2 = 20

. Rewriting the index equation in terms of base 2 yields

\frac{4x-2}{3} = -4x + 16

, which gives

x = \frac{25}{8}

and exponent

k = 3.5

. Adding

M^2

and

k

results in 23.5.

Step-by-Step Solution

1

Simplify the nested surd expression for M

M = 2\sqrt{5}, so M^2 = 20

Recognize that

7 \pm 2\sqrt{10} = (\sqrt{5} \pm \sqrt{2})^2

.

2

Convert both sides of the exponential equation to base 2

2^{\frac{4x-2}{3}} = 2^{-4x+16}

Apply laws of indices:

\sqrt[3]{4} = 2^{2/3}

and

\frac{1}{16} = 2^{-4}

.

3

Solve for x by equating the powers of 2

x = \frac{25}{8} = 3.125

Since bases are equal, the powers must be equal.

4

Determine the exponent value k

k = 3.5

Substitute x into the exponent expression

k = -4x + 16

.

5

Calculate the final combined expression M^2 + k

23.5

Add

M^2 = 20

and

k = 3.5

.

Key Concept

Nested radical simplification using binomial square expansion combined with solving exponential equations via prime base unification.

Question 8Question →

If $3^{x + 2} = 81$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is 2.

Rewriting 81 as

3^4

gives

3^{x+2} = 3^4

. Since the bases are equal, setting the exponents equal gives

x + 2 = 4

, which solves to

x = 2

.

Step-by-Step Solution

1

Express the right-hand side with a base of 3

81 = 3^4

Both sides must have the same base to apply the laws of indices.

2

Equate the exponents

x + 2 = 4

If

a^m = a^n

(where

a > 0

and

a \neq 1

), then

m = n

.

3

Solve for

x

x = 2

Subtracting 2 from both sides isolates

x

.

Key Concept

Equating exponents of equal bases

Question 9Question →

If $x = \sqrt{6 + 2\sqrt{5}} - \sqrt{6 - 2\sqrt{5}}$ and $\left(\frac{1}{27}\right)^{x - 5} = 9^{y + 1}$ , what is the value of $x + y$ ?

\frac{11}{2}

\frac{9}{2}

-\frac{7}{2}

10

Show answer & explanation

Answer: $\frac{11}{2}$

Answer

The value of

x + y

is

\frac{11}{2}

.

Simplifying

\sqrt{6 + 2\sqrt{5}}

yields

\sqrt{5} + 1

and

\sqrt{6 - 2\sqrt{5}}

yields

\sqrt{5} - 1

. Taking their difference gives

x = 2

. Substituting

x = 2

into the exponential equation gives

(1/27)^{-3} = 27^3 = (3^3)^3 = 3^9

. The right side is

9^{y+1} = (3^2)^{y+1} = 3^{2y+2}

. Equating exponents gives

2y + 2 = 9

, which leads to

y = 7/2

. Adding

x

and

y

gives

2 + 7/2 = 11/2

.

Step-by-Step Solution

1

Simplify the nested surds for

x

\sqrt{6 + 2\sqrt{5}} = \sqrt{5} + 1

and

\sqrt{6 - 2\sqrt{5}} = \sqrt{5} - 1

For a nested surd

\sqrt{a \pm 2\sqrt{b}}

, express

a

as

m+n

and

b

as

m \cdot n

. Here

5+1=6

and

5 \cdot 1 = 5

.

2

Compute the value of

x

x = (\sqrt{5} + 1) - (\sqrt{5} - 1) = 2

Subtracting the second simplified surd from the first cancels out

\sqrt{5}

and adds the integer terms.

3

Express the exponential equation using a common prime base

3

(3^{-3})^{2 - 5} = (3^2)^{y + 1} \implies (3^{-3})^{-3} = 3^{2y + 2} \implies 3^9 = 3^{2y + 2}

Rewriting

\frac{1}{27} = 3^{-3}

and

9 = 3^2

allows direct comparison of exponents.

4

Solve for

y

and evaluate

x + y

2y + 2 = 9 \implies y = \frac{7}{2}

, so

x + y = 2 + \frac{7}{2} = \frac{11}{2}

Equating exponents with identical bases and adding the resolved values of

x

and

y

.

Key Concept

Simplification of nested surds

\sqrt{a \pm 2\sqrt{b}}

combined with solving exponential equations using common prime bases.

Estimated Time:2m 0s

Question 10Question →

What is the simplified value of the expression $(256)^{0.16} \times (256)^{0.09}$ ?

4

16

64

1

Show answer & explanation

Answer: $4$

Answer

The simplified value of

(256)^{0.16} \times (256)^{0.09}

is

4

.

According to the basic laws of indices, multiplying terms with equal bases requires adding their exponents:

(256)^{0.16} \times (256)^{0.09} = (256)^{0.16 + 0.09} = (256)^{0.25}

. Expressing

0.25

as

\frac{1}{4}

converts the expression to the fourth root of

256

, which equals

4

because

4^4 = 256

.

Step-by-Step Solution

1

Apply the law of indices

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

(256)^{0.16 + 0.09} = (256)^{0.25}

When multiplying exponential terms with identical bases, add their powers.

2

Convert the decimal exponent to a simplified fraction

0.25 = \frac{25}{100} = \frac{1}{4}

Converting to a fraction simplifies root evaluation.

3

Express

256

as a base with power

4

256 = 4^4

Writing the base as

4^4

allows straightforward simplification with the exponent

\frac{1}{4}

.

4

Evaluate

(4^4)^{1/4}

4^{4 \times \frac{1}{4}} = 4^1 = 4

Apply the power rule

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

to arrive at the final simplified value.

Key Concept

Laws of Indices: Product Rule (

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

) and Power of a Power Rule (

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

)

Question 11Question →

If $x = \sqrt{11 + 4\sqrt{6}} - \sqrt{11 - 4\sqrt{6}}$ and $\left(\frac{1}{16}\right)^{x^2 - 15} = 64^{y + 1}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 1

Answer

The value of

y

is

1

.

Simplifying

\sqrt{11 \pm 4\sqrt{6}}

as

\sqrt{8} \pm \sqrt{3}

yields

x = 2\sqrt{3}

and

x^2 = 12

. Substituting

x^2

gives

(1/16)^{-3} = 16^3 = 2^{12}

. Matching exponents with

64^{y+1} = 2^{6y+6}

yields

12 = 6y + 6

, giving

y = 1

.

Step-by-Step Solution

1

Simplify the nested surds for

x

x = 2\sqrt{3}

Express

11 \pm 4\sqrt{6}

as

11 \pm 2\sqrt{24} = (\sqrt{8} \pm \sqrt{3})^2

, giving

\sqrt{11 + 4\sqrt{6}} - \sqrt{11 - 4\sqrt{6}} = (2\sqrt{2} + \sqrt{3}) - (2\sqrt{2} - \sqrt{3}) = 2\sqrt{3}

.

2

Evaluate

x^2

x^2 = 12

Squaring

x = 2\sqrt{3}

yields

x^2 = 4 \times 3 = 12

.

3

Substitute

x^2 = 12

into the exponent of the given equation

\left(\frac{1}{16}\right)^{-3} = 64^{y + 1}

The exponent

x^2 - 15

evaluates to

12 - 15 = -3

.

4

Convert both sides to a common prime base (base 2)

2^{12} = 2^{6y + 6}

\left(\frac{1}{16}\right)^{-3} = 16^3 = (2^4)^3 = 2^{12}

and

64^{y + 1} = (2^6)^{y + 1} = 2^{6y + 6}

.

5

Equate powers and solve for

y

y = 1

Since the bases are identical,

12 = 6y + 6 \implies 6y = 6 \implies y = 1

.

Key Concept

Nested surd evaluation via perfect square conversion combined with solving exponential equations using index rules.

Question 12Question →

If $p = \sqrt{12 + \sqrt{12 + \sqrt{12 + \dots}}}$ and $q = \sqrt{12 - \sqrt{12 - \sqrt{12 - \dots}}}$ , such that $2^{p+q-z} = \frac{1}{64}$ , what is the value of $z$ ?

1

5

13

-1

Show answer & explanation

Answer: $13$

Answer

The value of

z

is

13

.

Solving

p = \sqrt{12 + p}

yields the quadratic equation

p^2 - p - 12 = 0

, giving

p = 4

. Solving

q = \sqrt{12 - q}

yields

q^2 + q - 12 = 0

, giving

q = 3

. Adding

p

and

q

gives

p + q = 7

. Substituting this into

2^{p+q-z} = \frac{1}{64}

produces

2^{7-z} = 2^{-6}

. Equating the indices gives

7 - z = -6

, which solves to

z = 13

.

Step-by-Step Solution

1

Evaluate the first infinite nested surd

p

p = 4

Let

p = \sqrt{12 + p}

. Squaring both sides yields

p^2 = 12 + p \implies p^2 - p - 12 = 0

. Factoring gives

(p-4)(p+3) = 0

. Since

p > 0

,

p = 4

.

2

Evaluate the second infinite nested surd

q

q = 3

Let

q = \sqrt{12 - q}

. Squaring both sides yields

q^2 = 12 - q \implies q^2 + q - 12 = 0

. Factoring gives

(q+4)(q-3) = 0

. Since

q > 0

,

q = 3

.

3

Substitute

p

and

q

into the exponent expression

p + q = 7

, giving

2^{7-z} = \frac{1}{64}

Summing

p

and

q

gives

4 + 3 = 7

.

4

Solve the exponential equation for

z

z = 13

Express

\frac{1}{64}

as a base

2

power:

\frac{1}{64} = 2^{-6}

. Equating exponents gives

7 - z = -6 \implies z = 7 + 6 = 13

.

Key Concept

Evaluation of infinite nested surds and solving exponential equations using laws of indices

Estimated Time:2m 0s

Question 13Question →

What is the simplified value of the expression $\left(\frac{64}{125}\right)^{-\frac{2}{3}}$ ?

\frac{25}{16}

\frac{16}{25}

-\frac{25}{16}

\frac{5}{4}

Show answer & explanation

Answer: $\frac{25}{16}$

Answer

\frac{25}{16}

The expression

\left(\frac{64}{125}\right)^{-\frac{2}{3}}

is simplified by first converting the negative power to positive by inverting the fraction to

\left(\frac{125}{64}\right)^{\frac{2}{3}}

. Taking the cube root yields

\frac{5}{4}

, and squaring that result gives

\frac{25}{16}

.

Step-by-Step Solution

1

Apply the negative exponent rule

a^{-n} = \frac{1}{a^n}

to invert the fraction

\left(\frac{64}{125}\right)^{-\frac{2}{3}} = \left(\frac{125}{64}\right)^{\frac{2}{3}}

A negative exponent indicates taking the reciprocal of the base.

2

Express the base numbers as perfect cubes

125 = 5^3

and

64 = 4^3

, so

\frac{125}{64} = \left(\frac{5}{4}\right)^3

Rewriting bases into prime factors with power multiples simplifies fractional exponents.

3

Apply the power of a power rule

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

\left(\left(\frac{5}{4}\right)^3\right)^{\frac{2}{3}} = \left(\frac{5}{4}\right)^{3 \times \frac{2}{3}} = \left(\frac{5}{4}\right)^2

Multiplying the inner exponent

3

by the outer exponent

\frac{2}{3}

yields

2

.

4

Square the fraction

\left(\frac{5}{4}\right)^2 = \frac{25}{16}

Square both the numerator and the denominator to get the final numerical value.

Key Concept

Negative and Fractional Indices Rules

Question 14Question →

If $x = \sqrt{10 + 2\sqrt{21}} - \sqrt{10 - 2\sqrt{21}}$ and $9^{x^2 - k} = \left(\frac{1}{243}\right)^{k - 6}$ , find the value of $k$ .

Show answer & explanation

Answer: 2

Answer

The value of

k

is 2.

Simplifying

x

yields

x = (\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

, which gives

x^2 = 12

. Substituting

x^2 = 12

into the exponential equation gives

9^{12 - k} = (1/243)^{k - 6}

. Expressing both sides with prime base 3 gives

3^{2(12 - k)} = 3^{-5(k - 6)}

, which simplifies to

3^{24 - 2k} = 3^{30 - 5k}

. Equating the exponents gives

24 - 2k = 30 - 5k

, which solves to

k = 2

.

Step-by-Step Solution

1

Simplify the nested surds for

x

\sqrt{10 + 2\sqrt{21}} = \sqrt{7} + \sqrt{3}

and

\sqrt{10 - 2\sqrt{21}} = \sqrt{7} - \sqrt{3}

Nested surds of the form

\sqrt{a \pm 2\sqrt{b}}

simplify to

\sqrt{m} \pm \sqrt{n}

where

m + n = a

and

m \cdot n = b

.

2

Evaluate

x

and calculate

x^2

x = 2\sqrt{3}

and

x^2 = 12

Subtracting the two surds gives

(\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

. Squaring

2\sqrt{3}

gives

(2)^2 \times 3 = 12

.

3

Express both sides of the exponential equation with base 3

3^{24 - 2k} = 3^{30 - 5k}

Since

9 = 3^2

and

243 = 3^5

,

9^{12 - k} = (3^2)^{12 - k} = 3^{24 - 2k}

and

(1/243)^{k - 6} = (3^{-5})^{k - 6} = 3^{30 - 5k}

.

4

Equate the powers and solve for

k

k = 2

Since bases are equal (

3 > 0, 3 \neq 1

), setting exponents equal gives

24 - 2k = 30 - 5k \implies 3k = 6 \implies k = 2

.

Key Concept

Simplification of nested square root surds

\sqrt{a \pm 2\sqrt{b}}

combined with equating exponents using prime base factorization.

Question 15Question →

If $5^{x+1} + 5^{x-1} = 650$ , what is the value of $2^{2x - 1}$ ?

32

64

16

8

Show answer & explanation

Answer: $32$

Answer

32

Factoring out

5^{x-1}

yields

26 \cdot 5^{x-1} = 650

, which simplifies to

5^{x-1} = 25

, giving

x = 3

. Substituting

x = 3

into

2^{2x - 1}

gives

2^5 = 32

.

Step-by-Step Solution

1

Factor out common exponential terms from the given equation.

5^{x-1}(5^2 + 1) = 650 \implies 5^{x-1}(25 + 1) = 650 \implies 26 \cdot 5^{x-1} = 650

Use the product law of indices

5^{x+1} = 5^{x-1} \cdot 5^2

to rewrite the expression.

2

Solve for the unknown variable

x

.

5^{x-1} = \frac{650}{26} = 25 = 5^2 \implies x - 1 = 2 \implies x = 3

Since the bases are identical, equate the exponents.

3

Substitute

x = 3

into the expression

2^{2x - 1}

.

2^{2(3) - 1} = 2^{6 - 1} = 2^5 = 32

Evaluate the power after computing the exponent value.

Key Concept

Solving Exponential Equations using Laws of Indices

Question 16Question →

If $x = \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}}$ and $y = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}}$ , what is the value of $x^2 + y^2 + xy$ ?

142

143

144

145

Show answer & explanation

Answer: $143$

Answer

The value of

x^2 + y^2 + xy

is

143

.

Rationalizing both surds gives

x = 6 + \sqrt{35}

and

y = 6 - \sqrt{35}

. The sum

x + y = 12

and product

xy = 1

. Substituting into the algebraic identity

x^2 + y^2 + xy = (x + y)^2 - xy

gives

12^2 - 1 = 143

.

Step-by-Step Solution

1

Rationalize the denominator for

x

and

y

x = \frac{(\sqrt{7} + \sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{7 + 5 + 2\sqrt{35}}{2} = 6 + \sqrt{35}

, and similarly

y = 6 - \sqrt{35}

.

Eliminating radicals from denominators simplifies calculations.

2

Compute the sum

(x + y)

and product

xy

x + y = (6 + \sqrt{35}) + (6 - \sqrt{35}) = 12

, and

xy = (6 + \sqrt{35})(6 - \sqrt{35}) = 36 - 35 = 1

.

Using symmetric expressions simplifies evaluating degree 2 polynomials.

3

Express

x^2 + y^2 + xy

in terms of

(x + y)

and

xy

x^2 + y^2 + xy = (x + y)^2 - xy = 12^2 - 1 = 144 - 1 = 143

.

Applying the identity

x^2 + y^2 = (x + y)^2 - 2xy

gives

x^2 + y^2 + xy = (x + y)^2 - xy

.

Key Concept

Rationalization of surds and application of algebraic identities

Question 17Question →

If $\frac{3^{2m + 1} \cdot 9^{m - 2}}{27^{m - 1}} = 81$ , what is the value of $m$ ?

Show answer & explanation

Answer: 4

Answer

The value of

m

is 4.

By converting all terms in the equation to base 3 and applying index rules (

a^x \cdot a^y = a^{x+y}

and

\frac{a^x}{a^y} = a^{x-y}

), the expression simplifies directly to

3^m = 3^4

, leading to

m = 4

.

Step-by-Step Solution

1

Express all terms with a common base of 3

9^{m-2} = (3^2)^{m-2} = 3^{2m-4}

and

27^{m-1} = (3^3)^{m-1} = 3^{3m-3}

Rewriting powers with a single prime base allows for simplification using the laws of indices.

2

Simplify the numerator using the product rule of exponents

3^{2m+1} \cdot 3^{2m-4} = 3^{(2m+1) + (2m-4)} = 3^{4m-3}

According to the product rule

a^x \cdot a^y = a^{x+y}

, exponents with the same base are added during multiplication.

3

Apply the quotient rule of exponents to simplify the left-hand side

\frac{3^{4m-3}}{3^{3m-3}} = 3^{(4m-3) - (3m-3)} = 3^m

According to the quotient rule

\frac{a^x}{a^y} = a^{x-y}

, exponents are subtracted during division.

4

Express the right-hand side in base 3 and solve for

m

3^m = 81 = 3^4 \implies m = 4

Since the bases on both sides of the equation are equal, the exponents must also be equal.

Key Concept

Exponential Equations and Laws of Indices

Question 18Question →

If $\left(\sqrt{\frac{3}{5}}\right)^{x + 1} = \left(\frac{125}{27}\right)^{x - 1}$ , what is the value of $x$ ?

\frac{5}{7}

\frac{7}{5}

\frac{1}{2}

\frac{1}{3}

Show answer & explanation

Answer: $\frac{5}{7}$

Answer

\frac{5}{7}

By converting both sides of the equation to the common base

\frac{3}{5}

, the left side becomes

\left(\frac{3}{5}\right)^{\frac{x+1}{2}}

and the right side becomes

\left(\frac{3}{5}\right)^{-3(x-1)}

. Equating exponents gives

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

, which simplifies directly to

x = \frac{5}{7}

.

Step-by-Step Solution

1

Express the left side of the equation using fractional exponents

\left(\sqrt{\frac{3}{5}}\right)^{x+1} = \left(\left(\frac{3}{5}\right)^{\frac{1}{2}}\right)^{x+1} = \left(\frac{3}{5}\right)^{\frac{x+1}{2}}

The square root of a quantity corresponds to an exponent of

\frac{1}{2}

.

2

Express the right side with base

\frac{3}{5}

\left(\frac{125}{27}\right)^{x-1} = \left(\left(\frac{5}{3}\right)^3\right)^{x-1} = \left(\left(\frac{3}{5}\right)^{-3}\right)^{x-1} = \left(\frac{3}{5}\right)^{-3(x-1)}

Since

125 = 5^3

and

27 = 3^3

,

\frac{125}{27} = \left(\frac{5}{3}\right)^3 = \left(\frac{3}{5}\right)^{-3}

.

3

Equate the exponents since the bases are identical

\frac{x+1}{2} = -3(x-1) \implies \frac{x+1}{2} = 3 - 3x

If

a^m = a^n

for

a > 0

and

a \neq 1

, then

m = n

.

4

Solve the linear equation for

x

x + 1 = 2(3 - 3x) \implies x + 1 = 6 - 6x \implies 7x = 5 \implies x = \frac{5}{7}

Standard algebraic simplification to isolate

x

.

Key Concept

Laws of Indices and Rational Base Equivalence

Question 19Question →

If $x = 7 - 4\sqrt{3}$ , what is the value of $\sqrt{x} + \frac{1}{\sqrt{x}}$ ?

Show answer & explanation

Answer: 4

Answer

The value of

\sqrt{x} + \frac{1}{\sqrt{x}}

is 4.

Expressing

7 - 4\sqrt{3}

as

(2 - \sqrt{3})^2

allows taking the square root to get

\sqrt{x} = 2 - \sqrt{3}

. Rationalizing its reciprocal gives

\frac{1}{\sqrt{x}} = 2 + \sqrt{3}

. Adding these two values cancels the irrational component

\sqrt{3}

, leaving

2 + 2 = 4

.

Step-by-Step Solution

1

Simplify the nested surd

\sqrt{x} = \sqrt{7 - 4\sqrt{3}}

\sqrt{x} = 2 - \sqrt{3}

Rewrite

7 - 4\sqrt{3}

as

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

and take the principal square root.

2

Calculate the reciprocal

\frac{1}{\sqrt{x}}

by rationalizing the denominator

\frac{1}{\sqrt{x}} = 2 + \sqrt{3}

Multiply numerator and denominator of

\frac{1}{2 - \sqrt{3}}

by its conjugate

(2 + \sqrt{3})

.

3

Add

\sqrt{x}

and

\frac{1}{\sqrt{x}}

4

Sum

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

so that the radical terms cancel out.

Key Concept

Simplification of surds of the form

\sqrt{a \pm \sqrt{b}}

and rationalization using conjugates

Question 20Question →

If $2^{x+1} + 2^{x-1} = 320$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x

is 7.

Factoring out

2^{x-1}

from the expression gives

2^{x-1}(2^2 + 1) = 320

, which simplifies to

5 \cdot 2^{x-1} = 320

. Dividing by 5 yields

2^{x-1} = 64 = 2^6

. Equating exponents gives

x - 1 = 6

, so

x = 7

.

Step-by-Step Solution

1

Rewrite terms using exponent rules to factor out the common power

2^{x-1}

2^{x-1}(2^2 + 1) = 320

Using the product rule

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

, we have

2^{x+1} = 2^{x-1} \cdot 2^2

.

2

Simplify the numerical factor inside the parentheses

5 \cdot 2^{x-1} = 320

2^2 + 1 = 4 + 1 = 5

.

3

Isolate the exponential term by dividing by 5

2^{x-1} = 64

Dividing

320

by

5

gives

64

.

4

Write 64 as a power of 2 and solve for

x

2^{x-1} = 2^6 \implies x - 1 = 6 \implies x = 7

Equating exponents when bases are equal.

Key Concept

Solving Exponential Equations by Factoring Common Powers

Surds and Indices

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