Algebraic Expressions and Simplifications

Problems and Solutions

Problem 11

Simplify: p16x2+924x−16x2−9A\frac{p16x^2 + 9}{24x} - \frac{16x^2 - 9}{A}

The possible answers are:

A) 14x+3\frac{1}{4x+3}

B) Ω{14x+3,amp;if xlt;34 14x+3,amp;if xgt;34\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) <br>Ω{<br>−14x+3,amp;if xlt;34 14x+3,amp;if xgt;34<br><br><br>\Omega \begin{cases}<br>-\frac{1}{4x+3}, &amp; \text{if } x &lt; \frac{3}{4} \ \frac{1}{4x+3}, &amp; \text{if } x &gt; \frac{3}{4}<br>\end{cases}<br>

D) 14x+3\frac{1}{4x+3}

E) Cannot be simplified

Problem 12

If a=2a = \sqrt{2} and b=3b = \sqrt{3}, calculate the value of a2−2ab+b2+a2+2ab+b2\sqrt{a^2 - 2ab + b^2} + \sqrt{a^2 + 2ab + b^2}.

The possible answers are:

A) 8\sqrt{8}

B) 3⋅12\sqrt{3} \cdot 12

C) 18\sqrt{18}

D) 3⋅24\sqrt{3} \cdot 24

E) 27\sqrt{27}

Problem 1

Simplify: 3a−a2−3+3a+a2−33\sqrt{a} - \sqrt{a^2 - 3} + 3\sqrt{a} + \sqrt{a^2 - 3}

The possible answers are:

A) 1.5a1.5a

B) 3a3a

C) 2a2a

D) 2.5a2.5a

E) 2.4a2.4a

Problem 14

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

The possible answers are:

A) −0.05-0.05

B) 0.050.05

C) 0.50.5

D) −0.5-0.5

E) 0.0050.005

Problem 15

If a=4−1a = 4^{-1}, b=42ab = 4^{2a}, and c=4bc = 4^b, what is the value of acb\frac{ac}{b}?

The possible answers are:

A) 22

B) 44

C) 88

D) 11

E) 0.50.5

Problem 16

If a=1232+23!a = \frac{1}{2\sqrt[2]{3} + \sqrt[3]{2}!}, find the value of a2−1a−a2−1a\sqrt{a^2} - \frac{1}{a} - \sqrt{a^2} - \frac{1}{a}.

The possible answers are:

A) 14\frac{1}{4}

B) 34\frac{3}{4}

C) 12\frac{1}{2}

D) 18\frac{1}{8}

E) 58\frac{5}{8}

Problem 17

If x=56x = 5\sqrt{6} and y=65y = 6\sqrt{5}, calculate the value of x2+2xy+y2−x2−2xy+y2\sqrt{x^2 + 2xy + y^2} - \sqrt{x^2 - 2xy + y^2}.

The possible answers are:

A) 720\sqrt{720}

B) 700\sqrt{700}

C) 640\sqrt{640}

D) 600\sqrt{600}

E) 560\sqrt{560}

Problem 18

If a=5.2a = 5.2, find the value of a2−a−6(a+3)a2−4−a2+a−6(a−3)a2−4\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) 1.51.5

B) −2.5-2.5

C) −1.5-1.5

D) 2.42.4

E) −3.2-3.2

Problem 19

Simplify: 1a+2−a2+2a3+2a2!−1⋅a2−12+1a!−1⋅2a+2\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) 12\sqrt{\frac{1}{2}}

B) 22

C) −2-2

D) 1a2\frac{1}{a\sqrt{2}}

E) −a2-a\sqrt{2}

Problem 20

Simplify: 1+x+xxx−1!−1−x1−x\sqrt{1 + \frac{\sqrt{x} + x}{x\sqrt{x} - 1}!^{-1}} - \frac{x}{1 - \sqrt{x}}

The possible answers are:

A) x+1\sqrt{x} + 1

B) 11

C) x−1\sqrt{x} - 1

D) −1-1

E) x\sqrt{x}

Problem 21

Simplify: x+1−xx+x+x1x−x2+x\sqrt{x} + 1 - \frac{x\sqrt{x} + x + \sqrt{x}}{\frac{1}{\sqrt{x} - x^2 + x}}

The possible answers are:

A) 2x2x

B) 22

C) 11

D) 2x−12x - 1

E) −1-1

Problem 22

Simplify: 3a−a2−3+3a+a2−33\sqrt{a} - \sqrt{a^2 - 3} + 3\sqrt{a} + \sqrt{a^2 - 3}

The possible answers are:

A) 1.5a1.5a

B) 3a3a

C) 2.5a2.5a

D) 2a2a

E) 2.4a2.4a

Problem 23

Simplify: y−xy−xy+x+xxx+yy!⋅xx+yyy3\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) x+y\sqrt{x} + \sqrt{y}

B) x−y\sqrt{x} - \sqrt{y}

C) x\sqrt{x}

D) y\sqrt{y}

E) 1y2\frac{1}{y^2}

Problem 24

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

The possible answers are:

A) 2x−12\sqrt{x} - 1

B) 22

C) −2-2

D) −2x−1-2\sqrt{x} - 1

E) 44

Problem 25

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

The possible answers are:

A) c−1\sqrt{c} - 1

B) c−1c - 1

C) c+1c + 1

D) c+1\sqrt{c} + 1

E) 11

Problem 26

If x=45mx = \frac{4}{5}m, find the value of m+x+m−xm+x−m−x\frac{\sqrt{m + x} + \sqrt{m - x}}{\sqrt{m + x} - \sqrt{m - x}}.

The possible answers are:

A) 22

B) 2m2m

C) 44

D) −2-2

E) 4m4m

Problem 27

If x < 0, simplify: x2−12x+36−x2\sqrt{x^2 - 12x + 36} - \sqrt{x^2}

The possible answers are:

A) 66

B) −6-6

C) 6−2x6 - 2x

D) 2x−62x - 6

E) 88

Problem 28

Simplify: a⋅a+b2ba!−1+b⋅a+b2ab!−1a \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) 2ab2ab

B) abab

C) 4ab4ab

D) 12ab\frac{1}{2}ab

E) 14ab\frac{1}{4}ab

Problem 29

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

The possible answers are:

A) 11

B) −1-1

C) 0.50.5

D) 0.250.25

E) 22

N-th Degree Root. Rational Index

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

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

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

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

  4. abm=am⋅bma^{\sqrt[m]{b}} = \sqrt[m]{a^m \cdot b}

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

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

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

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

  9. −a2n+1=−a2n+1\sqrt[2n+1]{-a} = -\sqrt[2n+1]{a}

Problem (00-3-17)

Simplify: a−aaa23+a56+a+a23−aa3+a+2a\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 a6=x\sqrt[6]{a} = x, then a=x6a = x^6, a23=x4\sqrt[3]{a^2} = x^4, a=x3\sqrt{a} = x^3, a3=x2\sqrt[3]{a} = x^2. The expression becomes:

x6−x6⋅x3x4+x5+x6+x4−x6x2+x3+2x3=x6(1−x3)x4(1+x+x2)+x6x2(1+x)+2x3=x2(1−x)(1+x+x2)1+x+x2+x2(1−x)(1+x)1+x+2x3=x2(1−x)+x2(1−x)+2x3=x2−x3+x2−x3+2x3=2x2=2a3\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: 2a32\sqrt[3]{a}

Problem 1 (99-5-5)

Simplify: 27a+19a23−3a3+1−a3+1−27a−19a23+3a13+1\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 1243\frac{1}{243} in the form of a power with base 9.

A) 9−5/29^{-5/2}
B) 9−3/49^{-3/4}
C) 9−5/39^{-5/3}
D) 9−3/29^{-3/2}
E) 9−5/49^{-5/4}

Problem 3 (97-4-3)

Find the largest number among the given options:

A) 15\sqrt{15}
B) 653\sqrt[3]{65}
C) 814\sqrt[4]{81}
D) 44
E) 434\sqrt[4]{43}

Problem 4 (97-9-63)

Find the largest number:

A) 33
B) 263\sqrt[3]{26}
C) 10\sqrt{10}
D) 824\sqrt[4]{82}
E) 2425\sqrt[5]{242}

Problem 5 (98-5-7)

Calculate: 1523⋅3135−13\frac{15^{\frac{2}{3}} \cdot 3^{\frac{1}{3}}}{5^{-\frac{1}{3}}}

A) 4545
B) 1515
C) 55
D) 33
E) 3030

Problem 6 (98-9-27)

Simplify: a23⋅b23⋅((ab)−16)4(ab)−83\frac{a^{\frac{2}{3}} \cdot b^{\frac{2}{3}} \cdot ((ab)^{-\frac{1}{6}})^4}{(ab)^{-\frac{8}{3}}}

A) (ab)43(ab)^{\frac{4}{3}}
B) −(ab)43-(ab)^{\frac{4}{3}}
C) (ab)3(ab)^3
D) (ab)53(ab)^{\frac{5}{3}}
E) (ab)83(ab)^{\frac{8}{3}}

Problem 7 (98-4-9)

Calculate: (149)−12−(18)−13−6423A(\frac{1}{49})^{-\frac{1}{2}} - (\frac{1}{8})^{-\frac{1}{3}} - \frac{64^{\frac{2}{3}}}{A}

A) 34\frac{3}{4}
B) 516\frac{5}{16}
C) 25\frac{2}{5}
D) 47\frac{4}{7}
E) 56\frac{5}{6}

Problem 8 (99-7-9)

Calculate: 3013⋅32310−23\frac{30^{\frac{1}{3}} \cdot 3^{\frac{2}{3}}}{10^{-\frac{2}{3}}}

A) 1515
B) 2020
C) 6060
D) 4545
E) 3030

Problem 9 (00-10-5)

Calculate: 65⋅1/44−12+2−5⋅2−165 \cdot \sqrt[4]{1/4}^{-12} + 2^{-5} \cdot 2^{-1}

A) 12\frac{1}{2}
B) 22
C) 14\frac{1}{4}
D) 18\frac{1}{8}
E) 1A\frac{1}{A}

Problem 10 (00-3-6)

Calculate: 0.027−13−(−16)−2+25634−3−1+5.50.027^{-\frac{1}{3}} - (\frac{-1}{6})^{-2} + 256^{\frac{3}{4}} - 3^{-1} + 5.5

A) 3333
B) 32.9732.97
C) 3131
D) 3232
E) 31.9931.99

Problem 11 (98-9-18)

If n=81n = 81, what is the value of nn3\sqrt[3]{n\sqrt{n}}

A) 33
B) 66
C) 99
D) 44
E) 55

Problem 12 (98-11-55)

If a > 0, b > 0, and c < 0, which of the following is equal to a3b3c33\sqrt[3]{a^3b^3c^3}?

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