How to divide exponents - ISEE Upper Level Mathematics Achievement

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Question

Simplify: $\frac{x^{50}}{x^{10}}$

Answer

Apply the quotient of powers rule.

$\frac{x^{50}}{x^{10}} = x^{50-10} = x^{40}$

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Question

Simplify: $\frac{12x^{12}}{3x^{3}}$

Answer

Separate the fraction and apply the quotient of powers rule:

$\frac{12x^{12}}{3x^{3}} = \frac{12}{3} \cdot \frac{x^{12}}{x^{3}} = 4x^{12-3} = 4x^{9}$

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Question

Evaluate:

$\frac{120}{12^{0}}$

Answer

Any nonzero number raised to the power of 0 is equal to 1, so $12^{0} = 1$ , and

$\frac{120}{12^{0}} = \frac{120}{1} = 120$

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Question

Evaluate:

$\frac{25}{10 ^{0} -5^{0}}$

Answer

Any nonzero number raised to the power of 0 is equal to 1, so $5^{0} = 10 ^{0} =1$ . Therefore,

$\frac{25}{10 ^{0} -5^{0}} = \frac{25}{1-1} = \frac{25}{0}$

However, an expression with a denominator of 0 is undefined, so that is the correct choice.

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Question

Simplify:

$\frac{6x^6}{2x^3}$

Answer

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

We should separate the fraction and apply the quotient rule for exponents:

$\frac{6x^6}{2x^3}=\frac{6}{2}\times \frac{x^6}{x^3}=3\times x^{6-3}$

$\Rightarrow \frac{6x^6}{2x^3}=3x^3$

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Question

Simplify:

$\frac{4x^8}{2x^0}$

Answer

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

We also know that any non-zero number raised to the power of is .

$\frac{4x^8}{2x^0}=\frac{4}{2}\times \frac{x^8}{1}=2x^8$

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Question

Simplify:

$\frac{6x^4-6}{3x^2+3x^0}$

Answer

We know that any non-zero number raised to the power of is equal to .

$\frac{6x^4-6}{3x^2+3x^0}=\frac{6(x^4-1)}{3(x^2+x^0)}=\frac{6(x^4-1)}{3(x^2+1)}$

Now factor:

$\frac{6(x^4-1)}{3(x^2+1)}=\frac{6}{3}\frac{(x^2-1)(x^2+1)}{x^2+1}=2(x^2-1)$

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Question

Evaluate:

$\frac{4^0}{2\times 5^0-12^0}$

Answer

We know that any non-zero number raised to the power of is equal to .

$\frac{4^0}{2\times 5^0-12^0}=\frac{1}{2\times 1-1}=\frac{1}{2-1}=1$

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Question

Evaluate:

$\frac{y^{-2}}{y^{-5}}$

Answer

Based on the quotient rule for exponents, we know that, for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

$\frac{y^{-2}}{y^{-5}}=y^{-2-(-5)}=y^{-2+5}=y^{3}$

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Question

Simplify:

$(\frac{3x}{y^3})^{3}$

Answer

When a fraction is raised to a power, the numerator and the denominator are both raised to that power:

$(\frac{3x}{y^3})^{3}=\frac{(3x)^3}{(y^3)^3}$

The power rule says that in order to raise a power to a power, we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ ,

where and are real numbers.

$(\frac{3x}{y^3})^{3}=\frac{(3x)^3}{(y^3)^3}=\frac{(3^{3}\times x^3)}{y^{3\times 3}}=\frac{27x^3}{y^9}$

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Question

Evaluate:

$(\frac{x}{y^4})^{5}$

Answer

When a fraction is raised to a power, the numerator and the denominator are both raised to that power:

$(\frac{x}{y^4})^{5}=\frac{x^5}{(y^4)^5}$

The power rule says that in order to raise a power to a power, we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(\frac{x}{y^4})^{5}=\frac{x^5}{(y^4)^5}=\frac{x^5}{y^{4\times 5}}=\frac{x^5}{y^{20}}$

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Question

Simplify:

$\frac{x^{-4}}{y^{-8}}$

Answer

The Negative Exponent Rule says $a^{-n}=\frac{1}{a^n}$ .

$\frac{x^{-4}}{y^{-8}}=\frac{y^8}{x^4}$

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Question

Evaluate:

$(\frac{14x^2}{7x})^0$

Assume variables are non-zero.

Answer

We know that any non-zero number raised to the power of is equal to .

$(\frac{14x^2}{7x})^0=1$

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Question

Evaluate:

$(\frac{22x^3}{11y})^5$

Answer

When a fraction is raised to a power, the numerator and the denominator are both raised to that power. In addition, the power rule says that in order to raise a power to a power, multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(\frac{22x^3}{11y})^{5}=\frac{(22x^3)^{5}}{(11y)^5}=(\frac{22}{11})^5\times \frac{x^{3\times 5}}{y^5}=2^5\times \frac{x^{15}}{y^5}=\frac{32x^{15}}{y^5}$

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Question

Simplify:

$\frac{(2x^4y^{-1})^{-3}}{(3x^3y^2)^{-2}}$

Answer

Based on the negative exponent rule we have:

$a^{-n}=\frac{1}{a^n}$

Which says negative exponents in the numerator get moved to the denominator and become positive exponents. And negative exponents in the denominator get moved to the numerator and become positive exponents. So we can write:

$\frac{(2x^4y^{-1})^{-3}}{(3x^3y^2)^{-2}}=\frac{(3x^3y^2)^{2}}{(2x^4y^{-1})^{3}}$

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents.So we have:

$\frac{(3x^3y^2)^{2}}{(2x^4y^{-1})^{3}}=\frac{3^2\times x^{(3\times 2)}\times y^{(2\times 2)}}{2^3\times x^{(4\times 3)}\times y^{(-1\times 3)}}=\frac{9x^6y^4}{8x^{12}y^{-3}}$

Separate the equation:

$\Rightarrow \frac{9x^6y^4}{8x^{12}y^{-3}}=\frac{9}{8}\times \frac{x^6}{x^{12}}\times \frac{y^4}{y^{-3}}$

$=\frac{9}{8}\times x^{6-12}\times y^{4+3}=\frac{9}{8}x^{-6}y^7=\frac{9y^7}{8x^6}$

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Question

Evaluate:

$\frac{(11-7)^2}{(9-7)^3}$

Answer

First we need to subtract the numbers in parentheses. So we can write:

$\frac{(11-7)^2}{(9-7)^3}=\frac{4^2}{2^3}=\frac{4\times 4}{2\times 2\times 2}=\frac{16}8{}=2$

We can alternatively write:

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

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Question

Simplify:

$\frac{(x^2y^0)^2}{(x^{-1}y)^{4}}$

Answer

Based on the zero-exponent rule we have:

That means any non-zero number raised to the zero power is equal to . So we can write:

$\frac{(x^2y^0)^2}{(x^{-1}y)^{4}}=\frac{(x^2\times 1)^2}{(x^{-1}y)^{4}}=\frac{(x^2)^2}{(x^{-1}y)^{4}}$

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. In addition based on the negative exponent rule we have:

$a^{-n}=\frac{1}{a^n}$

Which says negative exponents in the numerator get moved to the denominator and become positive exponents. And negative exponents in the denominator get moved to the numerator and become positive exponents. So we can write:

$\frac{(x^2)^2}{(x^{-1}y)^{4}}=\frac{x^{2\times 2}}{(x^{-1\times 4})y^4}=\frac{x^4}{x^{-4}y^4}=\frac{x^{4+4}}{y^4}=\frac{x^8}{y^4}$

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Question

What is the value of the expression below?

$\frac{x^{6}}{x^{3}}$

Answer

When values having the same base are divided, their exponents are subtracted.

$\frac{x^{6}}{x^{3}}$

Therefore, the correct answer is

$x^{3}$

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Question

What is the value of the expression $\frac{(4.2\times 10^6)}{(7.0\times 10^{-2})}$ ?

Answer

When dealing with dividing scientific notation, you can divide the like terms separately. First, divide 4.2 by 7.0. This yields 0.6. Then divide the 10s. $10^6\div 10^{-2}=10^8$ because when you divide a term with the same base, you subtract exponents. This gives you a combined answer of $0.6\times 10^8$ . Since this is not in scientific notation, move the decimal over one place to the right so that it will be in the appropriate notation. This gives you $6.0\times 10^7$ .

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Question

What is a simplified version of the expression below?

$\frac{8x^{9}}{4x}$

Answer

When exponents are divided, the exponent in the denominator is subtracted from the exponent in the numerator. Given that the exponent in the numerator is , and the exponent in the denominator is , the resulting exponent will be because . Thus, will be raised to an exponent of .

Meanwhile, will be divided by , resulting in .

Therefore, the answer is $2x^{8}$ .

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