Algebraic Expressions and Simplifications

Problems and Solutions

Problem 11

Simplify: \frac{p16x^2 + 9}{24x} - \frac{16x^2 - 9}{A}

The possible answers are:

A) \frac{1}{4x+3}

B) \Omega \begin{cases} \frac{1}{4x+3}, & \text{if } x < \frac{3}{4} \ \frac{1}{4x+3}, & \text{if } x > \frac{3}{4} \end{cases}

C)
\Omega \begin{cases}
-\frac{1}{4x+3}, & \text{if } x < \frac{3}{4} \ \frac{1}{4x+3}, & \text{if } x > \frac{3}{4}
\end{cases}

D) \frac{1}{4x+3}

E) Cannot be simplified

Problem 12

If a = \sqrt{2} and b = \sqrt{3}, calculate the value of \sqrt{a^2 - 2ab + b^2} + \sqrt{a^2 + 2ab + b^2}.

The possible answers are:

A) \sqrt{8}

B) \sqrt{3} \cdot 12

C) \sqrt{18}

D) \sqrt{3} \cdot 24

E) \sqrt{27}

Problem 1

Simplify: 3\sqrt{a} - \sqrt{a^2 - 3} + 3\sqrt{a} + \sqrt{a^2 - 3}

The possible answers are:

A) 1.5a

B) 3a

C) 2a

D) 2.5a

E) 2.4a

Problem 14

If a = 0.0025, calculate the value of \frac{\sqrt{(a + 2)^2 - 8a}}{\sqrt{a} - \sqrt{\frac{2}{a}}}.

The possible answers are:

A) -0.05

B) 0.05

C) 0.5

D) -0.5

E) 0.005

Problem 15

If a = 4^{-1}, b = 4^{2a}, and c = 4^b, what is the value of \frac{ac}{b}?

The possible answers are:

A) 2

B) 4

C) 8

D) 1

E) 0.5

Problem 16

If a = \frac{1}{2\sqrt[2]{3} + \sqrt[3]{2}!}, find the value of \sqrt{a^2} - \frac{1}{a} - \sqrt{a^2} - \frac{1}{a}.

The possible answers are:

A) \frac{1}{4}

B) \frac{3}{4}

C) \frac{1}{2}

D) \frac{1}{8}

E) \frac{5}{8}

Problem 17

If x = 5\sqrt{6} and y = 6\sqrt{5}, calculate the value of \sqrt{x^2 + 2xy + y^2} - \sqrt{x^2 - 2xy + y^2}.

The possible answers are:

A) \sqrt{720}

B) \sqrt{700}

C) \sqrt{640}

D) \sqrt{600}

E) \sqrt{560}

Problem 18

If a = 5.2, find the value of \frac{a^2 - a - 6}{(a + 3)\sqrt{a^2 - 4}} - \frac{a^2 + a - 6}{(a - 3)\sqrt{a^2 - 4}}.

The possible answers are:

A) 1.5

B) -2.5

C) -1.5

D) 2.4

E) -3.2

Problem 19

Simplify: \sqrt{\frac{1}{a} + \sqrt{2 - \frac{a^2 + 2}{a^3} + \frac{2}{a^2}! - 1}} \cdot \sqrt{a^2 - \frac{1}{\sqrt{2 + \frac{1}{a}}! - 1}} \cdot \sqrt{\frac{2}{a} + \sqrt{2}}

The possible answers are:

A) \sqrt{\frac{1}{2}}

B) 2

C) -2

D) \frac{1}{a\sqrt{2}}

E) -a\sqrt{2}

Problem 20

Simplify: \sqrt{1 + \frac{\sqrt{x} + x}{x\sqrt{x} - 1}!^{-1}} - \frac{x}{1 - \sqrt{x}}

The possible answers are:

A) \sqrt{x} + 1

B) 1

C) \sqrt{x} - 1

D) -1

E) \sqrt{x}

Problem 21

Simplify: \sqrt{x} + 1 - \frac{x\sqrt{x} + x + \sqrt{x}}{\frac{1}{\sqrt{x} - x^2 + x}}

The possible answers are:

A) 2x

B) 2

C) 1

D) 2x - 1

E) -1

Problem 22

Simplify: 3\sqrt{a} - \sqrt{a^2 - 3} + 3\sqrt{a} + \sqrt{a^2 - 3}

The possible answers are:

A) 1.5a

B) 3a

C) 2.5a

D) 2a

E) 2.4a

Problem 23

Simplify: \sqrt{\frac{\sqrt{y} - \sqrt{x}}{y - \sqrt{xy} + x} + \frac{x}{x\sqrt{x} + y\sqrt{y}}!} \cdot \frac{x\sqrt{x} + y\sqrt{y}}{y^3}

The possible answers are:

A) \sqrt{x} + \sqrt{y}

B) \sqrt{x} - \sqrt{y}

C) \sqrt{x}

D) \sqrt{y}

E) \frac{1}{y^2}

Problem 24

Simplify: \frac{\sqrt{x} + 2\sqrt{x} - 1 + \sqrt{x} - 2\sqrt{x} - 1}{(1 \sum x \sum 2)}

The possible answers are:

A) 2\sqrt{x} - 1

B) 2

C) -2

D) -2\sqrt{x} - 1

E) 4

Problem 25

Reduce the fraction: \frac{c - 2\sqrt{c} + 1}{\sqrt{c} - 1}

The possible answers are:

A) \sqrt{c} - 1

B) c - 1

C) c + 1

D) \sqrt{c} + 1

E) 1

Problem 26

If x = \frac{4}{5}m, find the value of \frac{\sqrt{m + x} + \sqrt{m - x}}{\sqrt{m + x} - \sqrt{m - x}}.

The possible answers are:

A) 2

B) 2m

C) 4

D) -2

E) 4m

Problem 27

If x < 0, simplify: \sqrt{x^2 - 12x + 36} - \sqrt{x^2}

The possible answers are:

A) 6

B) -6

C) 6 - 2x

D) 2x - 6

E) 8

Problem 28

Simplify: a \cdot \sqrt{\frac{\sqrt{a} + \sqrt{b}}{2b\sqrt{a}}!^{-1}} + b \cdot \sqrt{\frac{\sqrt{a} + \sqrt{b}}{2a\sqrt{b}}!^{-1}}

The possible answers are:

A) 2ab

B) ab

C) 4ab

D) \frac{1}{2}ab

E) \frac{1}{4}ab

Problem 29

Simplify: \frac{\sqrt{x} + 4}{\sqrt{x} - 4} - 2 \frac{\sqrt{x} - 4}{\sqrt{x} + 4} + 2

The possible answers are:

A) 1

B) -1

C) 0.5

D) 0.25

E) 2

N-th Degree Root. Rational Index

For arbitrary a > 0, b > 0, and n, m \le N numbers:

  1. a^{\frac{n}{m}} = \sqrt[m]{a^n}

  2. \sqrt[m]{a \cdot b} = \sqrt[m]{a} \cdot \sqrt[m]{b}

  3. \sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}}

  4. a^{\sqrt[m]{b}} = \sqrt[m]{a^m \cdot b}

  5. \sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a}

  6. (\sqrt[n]{a})^m = \sqrt[n]{a^m}

  7. \sqrt[pn]{a} = \sqrt[nmp]{a}

  8. (\sqrt[pn]{a})^n = a

  9. \sqrt[2n+1]{-a} = -\sqrt[2n+1]{a}

Problem (00-3-17)

Simplify: \frac{a - a\sqrt{a}}{\sqrt[3]{a^2} + \sqrt[6]{a^5} + a + \sqrt[3]{a^2}} - \frac{a}{\sqrt[3]{a} + \sqrt{a} + 2\sqrt{a}}

Denoting \sqrt[6]{a} = x, then a = x^6, \sqrt[3]{a^2} = x^4, \sqrt{a} = x^3, \sqrt[3]{a} = x^2. The expression becomes:

\frac{x^6 - x^6 \cdot x^3}{x^4 + x^5 + x^6 + x^4} - \frac{x^6}{x^2 + x^3 + 2x^3} = \frac{x^6(1 - x^3)}{x^4(1 + x + x^2)} + \frac{x^6}{x^2(1 + x) + 2x^3} = \frac{x^2(1 - x)(1 + x + x^2)}{1 + x + x^2} + \frac{x^2(1 - x)(1 + x)}{1 + x + 2x^3} = x^2(1 - x) + x^2(1 - x) + 2x^3 = x^2 - x^3 + x^2 - x^3 + 2x^3 = 2x^2 = 2\sqrt[3]{a}

Answer: 2\sqrt[3]{a}

Problem 1 (99-5-5)

Simplify: \frac{27a + 1}{9\sqrt[3]{a^2} - 3\sqrt[3]{a} + 1} - \sqrt[3]{a} + 1 - \frac{27a - 1}{9\sqrt[3]{a^2} + 3a^{\frac{1}{3}} + 1}

Problem 2 (99-8-16)

Express \frac{1}{243} in the form of a power with base 9.

A) 9^{-5/2}
B) 9^{-3/4}
C) 9^{-5/3}
D) 9^{-3/2}
E) 9^{-5/4}

Problem 3 (97-4-3)

Find the largest number among the given options:

A) \sqrt{15}
B) \sqrt[3]{65}
C) \sqrt[4]{81}
D) 4
E) \sqrt[4]{43}

Problem 4 (97-9-63)

Find the largest number:

A) 3
B) \sqrt[3]{26}
C) \sqrt{10}
D) \sqrt[4]{82}
E) \sqrt[5]{242}

Problem 5 (98-5-7)

Calculate: \frac{15^{\frac{2}{3}} \cdot 3^{\frac{1}{3}}}{5^{-\frac{1}{3}}}

A) 45
B) 15
C) 5
D) 3
E) 30

Problem 6 (98-9-27)

Simplify: \frac{a^{\frac{2}{3}} \cdot b^{\frac{2}{3}} \cdot ((ab)^{-\frac{1}{6}})^4}{(ab)^{-\frac{8}{3}}}

A) (ab)^{\frac{4}{3}}
B) -(ab)^{\frac{4}{3}}
C) (ab)^3
D) (ab)^{\frac{5}{3}}
E) (ab)^{\frac{8}{3}}

Problem 7 (98-4-9)

Calculate: (\frac{1}{49})^{-\frac{1}{2}} - (\frac{1}{8})^{-\frac{1}{3}} - \frac{64^{\frac{2}{3}}}{A}

A) \frac{3}{4}
B) \frac{5}{16}
C) \frac{2}{5}
D) \frac{4}{7}
E) \frac{5}{6}

Problem 8 (99-7-9)

Calculate: \frac{30^{\frac{1}{3}} \cdot 3^{\frac{2}{3}}}{10^{-\frac{2}{3}}}

A) 15
B) 20
C) 60
D) 45
E) 30

Problem 9 (00-10-5)

Calculate: 65 \cdot \sqrt[4]{1/4}^{-12} + 2^{-5} \cdot 2^{-1}

A) \frac{1}{2}
B) 2
C) \frac{1}{4}
D) \frac{1}{8}
E) \frac{1}{A}

Problem 10 (00-3-6)

Calculate: 0.027^{-\frac{1}{3}} - (\frac{-1}{6})^{-2} + 256^{\frac{3}{4}} - 3^{-1} + 5.5

A) 33
B) 32.97
C) 31
D) 32
E) 31.99

Problem 11 (98-9-18)

If n = 81, what is the value of \sqrt[3]{n\sqrt{n}}

A) 3
B) 6
C) 9
D) 4
E) 5

Problem 12 (98-11-55)

If a > 0, b > 0, and c < 0, which of the following is equal to \sqrt[3]{a^3b^3c^3}?

A) a|bc|
B) -abc
C) ab|c|
D) |abc|
E) abc