Powers & Roots of Numbers - GMAT Quantitative

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Question

What is ?

Answer

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Question

What is $\sqrt{81}$ ?

Answer

$\sqrt{81}=9$

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Question

What is $\sqrt{225}$ ?

Answer

$\sqrt{225}=15$

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Question

Solve: $\small (\sqrt{5}+\sqrt{4})^2$

Answer

First, FOIL:

$\small (\sqrt5+\sqrt4)^2=(\sqrt5+\sqrt4)(\sqrt5+\sqrt4)=5+\sqrt20+\sqrt20+4$

$\small =9+2\sqrt20$

Factor out $\small \sqrt4$

$\small =9+4\sqrt5$

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Question

Solve: $\small \frac{10^9-10^7}{99}$

Answer

First factor. $\small \frac{10^9-10^7}{99}\ =\ \frac{(10^7)(10^2-1)}{99}$

Simplify. $\small \frac{(10^7)(10^2-1)}{99}\ =\ \frac{(10^7)(100-1)}{99}\ =\ \frac{(10^7)(99)}{99}\ =\ 10^7$

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Question

If $\small x^2=8$ , what is $\small x^4?$

Answer

$\small x^4=(x^2)^2=(8)^2=64$

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Question

If the side length of a cube is tripled, how does the volume of the cube change?

Answer

The equation for the volume of a cube is $L^{3}$ . If the length is tripled, it becomes $(3L)^{3}$ , and $3^{3}=27$ , so the volume increases by 27 times the original size.

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Question

Evaluate: $11^{100} - 11^{99}$

Answer

$11^{100} - 11^{99} = 11 \cdot 11^{99} - 1\cdot 11^{99} = (11-1) \cdot 11^{99} = 10 \cdot 11^{99}$

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Question

Solve: $\frac{(0.5)^{6}}{(0.5)^{9}}$

Answer

Solve $\dpi{100} \small : \frac{0.5^{6}}{0.5^{9}} = 0.5^{6-9}= 0.5^{-3}=\frac{1}{0.5^{3}}=\frac{1}{0.125} = 8$

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Question

$\frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} =$

Answer

$\frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} = \frac{y^{3-1}z^{4-3}}{x^{3-2}} = \frac{y^{2}z}{x}$

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Question

In the sequence 1, 3, 9, 27, 81, … , each term after the first is three times the previous term. What is the sum of the 9th and 10th terms in the sequence?

Answer

We can rewrite the sequence as $(3)^{0}$ , $(3)^{1}$ , $(3)^{2}$ , $(3)^{3}$ , $(3)^{4}$ , … ,

and we can see that the 9th term in the sequence is $(3)^{8}$ and the 10th term in the sequence is $(3)^{9}$ . Therefore, the sum of the 9th and 10th terms would be

$3^{8}+3^{9}=3^{8}\cdot \left ( 1+3 \right )=4\cdot 3^{8}$

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Question

Simplify this expression as much as possible:

$\sqrt{9x}+\sqrt{12x}+\sqrt{27x}$

Answer

$\sqrt{9x}+\sqrt{12x}+\sqrt{27x}$

$=\sqrt{9}\cdot \sqrt{x}+\sqrt{4}\cdot \sqrt{3x}+\sqrt{9}\cdot \sqrt{3x}$

$=3 \sqrt{x}+2 \sqrt{3x}+3 \sqrt{3x}$

$=3 \sqrt{x}+\left (2 +3 \right )\sqrt{3x}$

$=3 \sqrt{x}+5\sqrt{3x}$

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Question

Simplify $\sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$

Answer

This can either be done by brute force (slow) or by recognizing the properties of roots and exponents (fast). Roots are simply fractional exponents: $\sqrt{x}=x^{\frac{1}{2}}$ , $\sqrt[3]{x}=x^{\frac{1}{3}}$ , etc. so they can be done in any order.

So we see a cube root, we can immediately cancel that with the exponent of 3. taking us from here: $\sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$ to $\sqrt{\sqrt[4]{(4^2)^2}}$ . We now simplify to get $\sqrt{\sqrt[4]{(4^4)}} = \sqrt4 = 2$

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Question

$3^{5}\cdot 3^{2}=$

Answer

The two terms have the same base, 3. Therefore, we add the exponents to simplify:

$3^{5}\cdot 3^{2}=3^{5+2}=3^7$

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Question

Which of the following is equivalent to $\left (81 x \right )^{\frac{3}{4}}$ ?

Answer

$\left (81 x \right )^{\frac{3}{4}} =\left ( \sqrt[4]{ 81 x } \right) ^{3} = \left ( \sqrt[4]{ 81 } \right) ^{3} \cdot\left ( \sqrt[4]{ x } \right) ^{3}= 3^{3} \sqrt[4]{ x^{3} } = 27 \sqrt[4]{ x^{3} }$

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Question

What is ?

Answer

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Question

What is $\sqrt[3]{64}$ ?

Answer

$\sqrt[3]{64}= 4$

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Question

Find the remainder

$\frac{75^{75}}{38}$

Answer

Clearly, we aren't expected to calculate 75^75. The clue is that since we are supposed to divide by 38, let's see if we can rewrite the numerator so that we have a 38 in it.

For example: $75^{75}=(76-1)^{75}$ ,

and ,

so we have:

$75^{75}=(2*38-1)^{75}$

If we expanded this expression, all of the terms but one will have a in them, so all of the terms except for the last term, when divided by , will not leave a remainder.

The last term will be $(-1)^{75}$ , which is just .

Essentially, all we are asked to do is to figure out the remainder when is divided by .

Otherwise stated:

, where R is the remainder.

Clearly,

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Question

Give the area of the above kite.

Answer

$7^{2}+ 24^{2}= 49 + 576 = 625 = 25^{2}$ ,

so the lengths of the sides of each triangle comprise a Pythagorean triple. Therefore, each triangle is a right triangle with legs 7 and 24, and the kite is a composite of two such right triangles. Its area is therefore

$2 \times \frac{1}{2} \times 7 \times 24 = 168$

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Question

If $\sqrt[3]{4^{6}.8^{3}} = 2^{m}$ , what is the value of ?

Answer

We need to put the expression under the format of 2 to the power of m:

$\sqrt[3]{4^{6}.8^{3}} = \sqrt[3]{(2^{2})^{6}.(2^{3})^{3}} = \sqrt[3]{2^{12}.2^{9}}$

$\sqrt[3]{2^{21}}=(2^{21})^{\frac{1}{3}}=2^{\frac{21}{3}}=2^7$

Therefore the right answer is 7.

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