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GED Math : Complex Operations

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Example Questions

Example Question #471 : Numbers And Operations

Simplify: $(4x)^{-3} \cdot 8x$

Possible Answers:

$\frac{1}{8x^{4}}$

$\frac{1}{8x^{2}}$

$\frac{1}{2x^{4}}$

$\frac{1}{2x^{2}}$

Correct answer:

$\frac{1}{8x^{2}}$

Explanation:

$(4x)^{-3} \cdot 8x$

$=\frac{1}{(4x)^{3}} \cdot 8x$

$=\frac{8x}{4^{3}x^{3}}$

$=\frac{8x}{64x^{3}}$

$=\frac{8}{64} \cdot \frac{x}{x^{3}}$

$=\frac{1}{8} \cdot \frac{1}{x^{3-1}}$

$=\frac{1}{8} \cdot \frac{1}{x^{2}}$

$=\frac{1}{8x^{2}}$

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Example Question #1 : Exponents

Express 45,000,000 in scientific notation.

Possible Answers:

$4.5 \times 10^{8}$

$4.5 \times 10^{7}$

$45 \times 10^{7}$

$45 \times 10^{6}$

Correct answer:

$4.5 \times 10^{7}$

Explanation:

Write the number with a decimal point and without commas:

$45 \; 000 \; 000.$

Counting the number of places, move the decimal point to the left until it is after the first nonzero digit:

$4.5 \; 000 \; 000$

The resulting number is 4.5; the number of places the decimal point moved to the left was 7. In scientific notation, this number is $4.5 \times 10^{7}$ .

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Example Question #5 : Exponents

Express 0.000000038 in scientific notation.

Possible Answers:

$3.8 \times 10^{-7}$

$3.8 \times 10^{-8}$

$38 \times 10^{-8}$

$38 \times 10^{-7}$

Correct answer:

$3.8 \times 10^{-8}$

Explanation:

Write the number:

$0.000\; 000\; 038$

Counting the number of places, move the decimal point to the right until it is after the first nonzero digit:

$0 \; 000\; 000\; 03.8$

The resulting number is 3.8; the number of places the decimal point moved to the right was 8. In scientific notation, this number is $3.8 \times 10^{-8}$ .

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Example Question #2 : Exponents

Which of the following is not equal to ?

Do not use a calculator.

Possible Answers:

$\left ( -4 \right )^{3}$

$\left ( -2\right )^{6}$

$-4^{3}$

$-2^{6}$

Correct answer:

$\left ( -2\right )^{6}$

Explanation:

A positive number raised to any real power is positive. $-2^{6}$ and $-4^{3}$ are the opposites of powers of positive numbers; they are therefore the opposites of positive numbers, making both negative.

A negative number raised to an odd power, such as $\left ( -4 \right )^{3}$ , is negative.

A negative number raised to an even power, such as $\left ( -2\right )^{6}$ , is positive.

$\left ( -2\right )^{6}$ is the only positive number among the choices and can be deduced as the correct choice.

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Example Question #472 : Numbers And Operations

Without using a calculator, evaluate $0.4 ^{-3}$ .

Possible Answers:

-1.2

-0.064

0.83

15.625

Correct answer:

15.625

Explanation:

For any nonzero and for any , $a^{-m} =\left ( \frac{1}{a} \right )^{m}$ . Equivalently, any number raised to a negative power is equal to the reciprocal of that number raised to the power of the absolute value of the exponent.

The reciprocal of a number is the quotient of 1 and the number, so the reciprocal of 0.4 is

$1 \div 0.4 = 2.5$ .

Therefore,

$0.4 ^{-3} = 2.5 ^{3} = 15.625$ , which is the correct response.

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Example Question #7 : Exponents

Evaluate without using a calculator:

$7 ^{-3}$

Possible Answers:

$-\frac{1}{21}$

$\frac{1}{343}$

Correct answer:

$\frac{1}{343}$

Explanation:

$7 ^{-3} = \frac{1}{7^{3}} = \frac{1}{7 \times 7 \times 7} =\frac{1}{343}$

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Example Question #8 : Exponents

Simplify the expression:

$10 x^{-3}$

Possible Answers:

$\frac{10}{x^{3}}$

$-\frac{1,000}{x^{3}}$

$-30x^{3}$

$\frac{1}{1,000x^{3}}$

Correct answer:

$\frac{10}{x^{3}}$

Explanation:

A number or variable can be raised to a negative power by taking the reciprocal of the same number raised to the power of the absolute value of the exponent.

$10 x^{-3} = 10 \cdot \frac{1}{x^{ 3}} = \frac{10 }{1}\cdot \frac{1}{x^{ 3}} = \frac{10}{x^{ 3}}$

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Example Question #4 : Exponents

Without using a calculator, evaluate:

$\left (\frac{4}{5} \right )^{-3}$

Possible Answers:

$\frac{125}{64}$

$-\frac{64}{125}$

$\frac{15}{4}$

$-\frac{12}{5}$

Correct answer:

$\frac{125}{64}$

Explanation:

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

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Example Question #11 : Exponents

Order from least to greatest:

$7^{-2}, 5 ^{-2}, 2 ^{-6}$

Do not use a calculator.

Possible Answers:

$2 ^{-6}, 5 ^{-2} , 7^{-2}$

$7^{-2}, 2 ^{-6},5 ^{-2}$

$2 ^{-6}, 7^{-2}, 5 ^{-2}$

$5 ^{-2} , 2 ^{-6}, 7^{-2}$

Correct answer:

$2 ^{-6}, 7^{-2}, 5 ^{-2}$

Explanation:

For any nonzero and for any , $a^{-m} = \frac{1}{a^{m}}$ .

$7^{-2} = \frac{1}{7^{2}} = \frac{1}{7 \times 7 } = \frac{1}{49}$

$5 ^{-2} = \frac{1}{5 ^{2}} = \frac{1}{5 \times 5} = \frac{1}{25}$

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

Since 25 <49 < 64 ,

we can reverse the order when taking their reciprocals:

$\frac{1}{64} < \frac{1}{49}<\frac{1}{25}$

That is, $2 ^{-6}< 7^{-2}< 5 ^{-2}$ .

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Example Question #11 : Exponents

Order from least to greatest:

$\left (-5 \right )^{-3}, \left (-4 \right )^{-3}, \left ( -2\right ) ^{-7}$

Do not use a calculator.

Possible Answers:

$\left ( -2\right ) ^{-7} , \left (-5 \right )^{-3}, \left (-4 \right )^{-3}$

$\left ( -2\right ) ^{-7}, \left (-4 \right )^{-3} , \left (-5 \right )^{-3}$

$\left (-4 \right )^{-3} , \left (-5 \right )^{-3} , \left ( -2\right ) ^{-7}$

$\left (-5 \right )^{-3} , \left (-4 \right )^{-3} , \left ( -2\right ) ^{-7}$

Correct answer:

$\left (-4 \right )^{-3} , \left (-5 \right )^{-3} , \left ( -2\right ) ^{-7}$

Explanation:

For any nonzero and for any , $a^{-m} = \frac{1}{a^{m}}$ .

Also, any negative number raised to the power of an odd number is equal to the opposite of the same power of its absolute value.

Combine these concepts:

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

$\left (-4 \right )^{-3} = \frac{1 }{\left (-4 \right )^{3}} = \frac{1}{- (4^{3})} = -\frac{1}{64}$

$\left ( -2\right ) ^{-7} = \frac{1}{\left ( -2\right )^{7}} = \frac{1}{- \left ( 2 ^{7}\right )} = - \frac{1}{128}$

Since 64< 125 < 128 ,

the reciprocals are in reverse order of this:

$\frac{1}{64}> \frac{1}{125 }>\frac{1}{128}$

Their opposites reverse again:

$-\frac{1}{64}<- \frac{1}{125 }<-\frac{1}{128}$

Or, equivalently,

$\left (-4 \right )^{-3} < \left (-5 \right )^{-3} < \left ( -2\right ) ^{-7}$ .

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GED Math : Complex Operations

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #471 : Numbers And Operations

Example Question #1 : Exponents

Example Question #5 : Exponents

Example Question #2 : Exponents

Example Question #472 : Numbers And Operations

Example Question #7 : Exponents

Example Question #8 : Exponents

Example Question #4 : Exponents

Example Question #11 : Exponents

Example Question #11 : Exponents

All GED Math Resources

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