Exponents - GED Math

Card 0 of 275

Question

Solve: $\small 8^4=$

Answer

$\small 8^4=8\times8\times8\times8=4,096$

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Question

Write the following expression in expanded form:

$\small n^4=$

Answer

An exponent indicates the amount of times a number (or variable) should be multiplied by itself. For example, $\small 2^3=2\times2\times2=8$ .

In this instance, $\small n^4=n\times n\times n\times n$ .

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Question

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

Answer

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

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

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

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

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

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

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

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

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Question

Express 45,000,000 in scientific notation.

Answer

Write the number with a decimal point and without commas:

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

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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Question

Express 0.000000038 in scientific notation.

Answer

Write the number:

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

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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Question

Which of the following is not equal to ?

Do not use a calculator.

Answer

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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Question

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

Answer

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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Question

Order from least to greatest:

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

Do not use a calculator.

Answer

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 ,

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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Question

Order from least to greatest:

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

Do not use a calculator.

Answer

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 ,

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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Question

Without using a calculator, evaluate:

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

Answer

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. Apply this power, along with the power of a quotient property:

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

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Question

Evaluate without using a calculator:

$7 ^{-3}$

Answer

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

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Question

Simplify the expression:

$10 x^{-3}$

Answer

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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Question

Solve: $2^2\div 4^{3}$

Answer

Evaluate each term first.

Divide 4 with 64 and reduce.

$\frac{4}{64} = \frac{(4\times 1)}{(4\times 16)}$

The answer is: $\frac{1}{16}$

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Question

Simplify: $\frac{2^3}{6^2}$

Answer

Do not subtract the exponents or divide the bases! We will need to compute the numerator and denominator first.

$\frac{2^3}{6^2} = \frac{(2)(2)(2)}{(6)(6)} = \frac{8}{36}$

Reduce this fraction.

The answer is: $\frac{2}{9}$

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Question

Divide the following:

$m^6 \div m^2$

Answer

To divide variables with exponents, we will use the following formula:

$x^a \div x^b = x^{a-b}$

So, we get

$m^6 \div m^2 = m^{6-2}$

$m^6 \div m^2 = m^4$

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Question

Multiply the following:

$h^7 \cdot h^2$

Answer

To multiply variables with exponents, we will use the following formula:

$x^a \cdot x^b = x^{a+b}$

So we get

$h^7 \cdot h^2 = h^{7+2}$

$h^7 \cdot h^2 = h^9$

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Question

Divide the following:

$a^{12} \div a^4$

Answer

To divide variables with exponents, we will use the following formula:

$x^a \div x^b = x^{a-b}$

Now, let’s divide. We get

$a^{12} \div a^4 = a^{12-4}$

$a^{12} \div a^4 = a^8$

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Question

Multiply the following:

$x^9 \cdot x^2$

Answer

To multiply variables with exponents, we will use the following formula:

$x^a \cdot x^b = x^{a+b}$

Now, we will multiply. We get

$x^9 \cdot x^2 = x^{9 + 2}$

$x^9 \cdot x^2 = x^{11}$

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Question

Divide: $\frac{(-2)^3}{(-4)^{-1}}$

Answer

Simplify the numerator.

Simplify the denominator.

$(-4)^{-1} = \frac{1}{(-4)^1} =-\frac{1}{4}$

Divide the terms.

$\frac{(-2)^3}{(-4)^{-1}} = \frac{-8}{-\frac{1}{4}} = -8\div- \frac{1}{4}$

Take the reciprocal of the second term and change the sign to a multiplication.

$-8\div- \frac{1}{4} = -8\times -4 = 32$

The answer is:

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Question

Divide the following:

$n^8 \div n^2$

Answer

To divide variables with exponents, we will use the following formula:

$x^a \div x^b = x^{a-b}$

So, we get

$n^8 \div n^2 = n^{8-2}$

$n^8 \div n^2 = n^6$

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