How to divide exponents - ACT Math

Card 0 of 16

Question

Simplify the following:

$\frac{x^7}{x^3}$

Answer

These exponents have the same base, x, so they can be divided. To divide them, you take the exponent value in the numerator (the top exponent) and subtract the exponent value of the denominator (the bottom exponent). Here that means we take 7 – 3 so our answer is x4.

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Question

What is the value of m where:

$64^{\frac{n}{12}}=(m)4^{(\frac{1+m}{m+n})}$

Answer

If n=4, then 64(4/12)=64(1/3)=4. Then, 4=m4(1+m)/(m+4). If 2 is substituted for m, then 4=24(1+2)/(2+4)=241/2=2√4=22=4.

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Question

If , then $\frac{(c^3)^2}{c^2c^4}=?$

Answer

Start by simplifying the numerator and denominator separately. In the numerator, (c3)2 is equal to c6. In the denominator, c2 * c4 equals c6 as well. Dividing the numerator by the denominator, c6/c6, gives an answer of 1, because the numerator and the denominator are the equivalent.

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Question

If $a ot\equiv0$ , which of the following is equal to $\frac{(a^6*a^2)^3}{a^6}$ ?

Answer

The numerator is simplified to (by adding the exponents), then cube the result. a24/a6 can then be simplified to $a^{18}$ .

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Question

Simplify

Answer

When working with polynomials, dividing is the same as multiplying by the reciprocal. After multiplying, simplify. The correct answer for division is

and the correct answer for multiplication is

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Question

Simplify

$\dpi{100} \small \frac{20x^{4}y^{-3}z^{2}}{5z^{-1}y^{2}x^{2}}=$

Answer

Divide the coefficients and subtract the exponents.

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Question

Which of the following is equal to the expression , where

xyz ≠ 0?

Answer

(xy)4 can be rewritten as x4y4 and z0 = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.

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Question

$\dpi{100} \small \frac{7^{10}-7^{8}}{7^{9}-7^{7}}=$

Answer

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is .

$\frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}$

Now, we can cancel out the from the numerator and denominator and continue simplifying the expression.

$\frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}=\frac{7^3-7^1}{7^2-1}=\frac{343-7}{49-1}=\frac{336}{48}=7$

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Question

Simplify:

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

Answer

To simply exponents in a fraction, subtract the exponent for each variable in the denominator from the exponent in the numerator. This will leave you with

$\small x^{-1}y^{-1}z^2$ or $\small \frac{z^2}{xy}$

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Question

Simplify:

$\frac{(2x^{3}y)(4x^{5}y^{4}z)}{x^{2}y}$

Answer

$a^{m} \cdot a^{n} = a^{m+n}$

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

Use rule for multiplying exponents to simplify the numerator. $\frac{8x^{8}y^{5}z}{x^{2}y}$

Use rule for dividing exponents to simplify.

$\frac{x^{8}}{x^{2}}=x^{8-2}=x^{6}$

$\frac{y^{5}}{y^{1}} = y^{5-1} = y^{4}$

$8x^{6}y^{4}z$

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Question

Simplify:

$\left (\frac{18a^3b^3}{3a^{-1}b^{-2}} \right )^2$

Answer

Simplify:

$\left (\frac{18a^3b^3}{3a^{-1}b^{-2}} \right )^2$

Step 1: Simplify the fraction. When dividing exponents subtract the exponents on the bottom from the exponents on the top.

$\left (\frac{18a^3b^3}{3a^{-1}b^{-2}} \right )^2 =\left ( 6a^4b^5 \right )^2$

Step 2: Distribute the exponent. When raising an exponent to a power, multiply them together.

$(6a^4b^5)^2 = 36a^8b^{10}$

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Question

Simplify the following

$\frac{4x^5y^-^4z^4}{2x^-^2y^-^2z^5}$ .

Answer

Start by remembering that you "flip" negative exponents over the division bar, thus moving from the top to the bottom and vice-versa. (There are other ways to do this as well, though most students understand this way most easily.)

$\frac{4x^5x^2y^2z^4}{2y^4z^5}$

Next, you just eliminate common factors and combine on the numerator. First combine the s:

$\frac{4x^7y^2z^4}{2y^4z^5}$

Cancel out the numeric coefficient:

$\frac{2x^7y^2z^4}{y^4z^5}$

Now eliminate s and s:

$\frac{2x^7}{y^2z}$

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Question

If $x^{\frac{5}{2}} = 1024$ , what is ?

Answer

Using the properties of exponents, we can raise both sides to a reciprocal of the exponent of to find the value we need. Specifically...

$x^{\frac{5}{2}} = 1024$

$(x^{\frac{5}{2}})^{\frac{2}{5}} = 1024^{\frac{2}{5}}$

$x = 1024^{\frac{2}{5}} = (\sqrt[5]{1024})^{2} = 4^2 = 16$

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Question

$64^{\frac{4}{3}}$ can be written as which of the following?

A. $\sqrt[3]{64^4}$

B. $(\sqrt[3]{64})^{4}$

C.

Answer

C is computing the exponent, while A and B are equivalent due to properties of fractional exponents.

Remember that...

$a^{\frac{b}{c}} = \sqrt[c]{a^b} = (\sqrt[c]{a})^b$

$64^{\frac{4}{3}}=$ $\sqrt[3]{64^4}=$ $(\sqrt[3]{64})^{4}=$

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Question

Simplify:

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

Answer

When exponents with the same base are being divided, you may substract the exponent in the denominator from the exponent in the numerator to create a new exponent. In this case, you would subtract from , yielding as the new exponent. Keeping the same base, the answer becomes .

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Question

What is the simplified form for the following expression?

$\frac{(2xz^2)^3}{(4zx^2)^2}$

Answer

Break up by variable.

$\frac{2^3}{4^2}= \frac{1}{2}$

$\frac{x^3}{x^4}= \frac{1}{x}$

$\frac{z^6}{z^2} = z^4$

Therefore the simplified form becomes,

$\frac{(2xz^2)^3}{(4zx^2)^2}=\frac{z^4}{2x}$ .

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