Multiplying and Dividing Exponents - Algebra II

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Question

Simplify this expression:

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

Answer

When different powers of the same variable are multiplied, the exponents are added. When different powers of the same variable are divided, the exponents are subtracted. So, as an example:

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

For the above problem,

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

Therefore, the expression simplifies to:

$\left ( \frac{1}{x} \right ) \left ( \frac{1}{y} \right ) z^{2} = \frac{z^{2}}{xy}$

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Question

Simplify.

$x^{-6}x^3x^9$

Answer

Put the negative exponent on the bottom so that you have $\frac{x^{12}}{x^{6}}$ which simplifies further to $x^{6}$ .

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Question

Simplify the rational expression.

$\frac{12x^2y^4z^6}{6xz^{12}}$

Answer

$\frac{12x^2y^4z^6}{6xz^{12}}$

To simplify variables with exponents through division, you must subtract the exponent in the denominator from the numerator.

$\frac{12}{6}(x^{2-1})(y^{4-0})(z^{6-12})=2xy^4z^{-6}=\frac{2xy^4}{z^6}$

Remember that negative exponents will eventually be moved back to the denominator.

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Question

Simplify $\frac{a^{4}b^{2}}{6a^{3}b}\div \frac{2b^{5}}{12a^{7}}$

Answer

Rewrite so that you are multiplying the reciprocal of the second fraction:

$\frac{a^{4}b^{2}}{6a^{3}b}\times \frac{12a^{7}}{2b^{5}}$

You can then simplify using rules of exponents:

$\frac{12a^{11}b^2}{12a^3b^6}=\frac{a^8}{b^4}$

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Question

Simplify the expression.

Answer

Rearrange the expression so that the $\small y$ and $\small z$ variables of different powers are right next to each other.

When multiplying the same variable with different exponents, it is the same as adding the exponents: $a^ba^c=a^{b+c}$ . Taking advantage of this rule, the problem can be rewritten.

$y^{2+4+6}z^{3+5}$

$y^{6+6}z^8$

$y^{12}z^8$

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Question

Simplify:

$\frac{9ab^2c}{4c^3} \times \frac{a^2b^2}{3a^3c}$

Answer

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators, by dividing 9 and 3 each by 3:

$\frac{9ab^2c}{4c^3} \times \frac{a^2b^2}{3a^3c} = \frac{3ab^2c}{4c^3} \times \frac{a^2b^2}{a^3c}$

Next you can multiply the two numerators, and multiply the two denominators. Remember that when multiplying like variables with exponents, you add the exponents together:

$\frac{3ab^2c}{4c^3} \times \frac{a^2b^2}{a^3c} = \frac{3a^3b^4c}{4a^3c^4}$

If a variable shows up in both the numerator and denominator, you can simplify by subtracting the numerator's exponent by the denominator's exponent. If you end up with a negative exponent in the numerator, you can move the variable and exponent to the denominator to make it positive:

$\frac{3b^4}{4c^3}$

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Question

Simplify the following:

$\frac{10x^2y^3z}{7xz^3} \times \frac{21xyz^2}{5y}$

Answer

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators, by dividing 10 and 5 each by 5 and dividing 21 and 7 each by 7:

$\frac{10x^2y^3z}{7xz^3} \times \frac{21xyz^2}{5y} = \frac{2x^2y^3z}{xz^3} \times \frac{3xyz^2}{y}$

Next you can multiply the two numerators, and multiply the two denominators. Remember that when multiplying like variables with exponents, you add the exponents together:

$\frac{2x^2y^3z}{xz^3} \times \frac{3xyz^2}{y} = \frac{6x^3y^4z^3}{xyz^3}$

If a variable shows up in both the numerator and denominator, you can simplify by subtracting the numerator's exponent by the denominator's exponent:

$\frac{6x^3y^4z^3}{xyz^3} = 6x^2y^3$

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Question

Simplify the following:

$\frac{2a^3bc}{3a^2bc^4} \times \frac{15ab^3c^2}{5a^3b^5c^2}$

Answer

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators. The two coefficients in the denominators multiply up to 15, allowing you to divide and cancel those two coefficients with the 15 in the numerator:

$\frac{2a^3bc}{3a^2bc^4} \times \frac{15ab^3c^2}{5a^3b^5c^2} = \frac{2a^3bc}{a^2bc^4} \times \frac{ab^3c^2}{a^3b^5c^2}$

Next you can multiply the two numerators, and multiply the two denominators. Remember that when multiplying like variables with exponents, you add the exponents together:

$\frac{2a^3bc}{a^2bc^4} \times \frac{ab^3c^2}{a^3b^5c^2} = \frac{2a^4b^4c^3}{a^5b^6c^6}$

If a variable shows up in both the numerator and denominator, you can simplify by subtracting the numerator's exponent by the denominator's exponent. If you end up with a negative exponent in the numerator, you can move the variable and exponent to the denominator to make it positive:

$\frac{2a^4b^4c^3}{a^5b^6c^6} = \frac{2}{ab^2c^3}$

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Question

Simplify:

$\frac{30a^{2}b}{c}*\frac{bc^{2}}{5c}$

Answer

Combine like terms in the numerator and the denominator. Use the rules of exponents to combine and in the numerator and and in the denominator. Remember that $x^{a}x^{b}=x^{a+b}$ Then, divide 30 by 5 (the GCF).

$\frac{30a^{2}bbc^{2}}{5cc}=\frac{30a^{2}b^{2}c^{2}}{5c^{2}}=\frac{6a^{2}b^{2}c^{2}}{c^{2}}$

The GCF rule can also be used to remove $c^{2}$ from the numerator and the deonominator. $c^{2}$ goes into $c^{2}$ once.

$\frac{6a^{2}b^{2}c^{2}}{c^{2}}=6a^{2}b^{2}$

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Question

Evaluate the following expression

$(5^{0})^{2}$

Answer

$(5^{0})^{2}=(1)^{2}=1$

We can also solve this problem using a different apporach

$(5^{0})^{2}=5^{0\times2}=5^{0}=1$

Remember that any number raised to the 0th power equals 1

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Question

Simplify the following expression

$((yz)^2)^{3}$

Answer

$((yz)^{2})^{3}=(yz)^{6}=y^{6}z^{6}$

Alternatively,

$((yz)^{2})^{3}=(y^{2}z^{2})^3=y^{2\times3}z^{2\times3}=y^{6}z^{6}$

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Question

Evaluate the following expression

$\left (\frac{10^{3}}{10^{0}} \right )^{2}$

Answer

$\left (\frac{10^{3}}{10^{0}} \right )^{2}=\frac{10^{3\times2}}{10^{0\times2}}=\frac{10^{6}}{10^{0}}=\frac{1000000}{1}=1,000,000$

Alternatively,

$\left (\frac{10^{3}}{10^{0}} \right )^{2}=\left (10^{3-0} \right )^{2}=\left (10^{3} \right )^{2}=10^{3\times 2}=10^{6}=1,000,000$

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Question

Simplify the following expression

$((6^2)^{0})^{3}$

Answer

$((6^2)^{0})^{3}=6^{2\times 0\times 3}=6^0=1$

Remember that any number raised to the 0th power equals 1

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Question

Simplify:

$\frac{4a^{-5}b^{3}c^{2}}{12a^{-7}b^{-5}c^{-2}}$

Answer

$\frac{4a^{-5}b^{3}c^{2}}{12a^{-7}b^{-5}c^{-2}}$

Step 1: Simplify the exponents using the division of exponents rule (subtract exponents in demoninator from exponents in numerator).

$\frac{4a^{-5-(-7)}b^{3-(-5)}c^{2-(-2)}}{12}$

$\frac{4a^{2}b^{8}c^{4}}{12}$

Step 2: Reduce the fraction

$\frac{a^{2}b^{8}c^{4}}{3}$

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Question

Simplify:

$\frac{q^{7}r^{5}s^{16}}{q^{2}r^{3}s}$

Answer

Follow the division of exponents rule. Subract the exponents in the denominator from the exponents in the numerator.

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Question

Rewrite using a single exponent.

$\frac{(6x)^{8}}{(6x)^{3}}$

Answer

Based on the property for dividing exponents:

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

In this problem, a is equal to , so

$\frac{(6x)^{8}}{(6x)^{3}}=(6x)^{8-3}=(6x)^{5}$

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Question

Simplify:

$\frac{7a^{-7}b^{-8}c^{4}}{14a^{5}b^{-9}c{}}$

Answer

Simplify:

$\frac{7a^{-7}b^{-8}c^{4}}{14a^{5}b^{-9}c{}}$

Step 1: Use the division of exponents rule, and subtract the exponents in the denominator from the exponents in the numerator

$\frac{7a^{-12}bc^3}{14}$

Step 2: Move negative exponents in the numerator to the denominator

$\frac{7bc^3}{14a^{12}}$

Step 3: Simplify

$\frac{bc^3}{2a^{12}}$

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Question

Simplify:

$\frac{x^{6}zy^{-9}}{x^{-4}z^2y^3}$

Answer

Step 1: Use the division of exponents rule. Subtract the exponents in the numerator from the exponents in the denominator.

${x^{6+4}z^{1-2}y^{-9-3}$

$x^{10}z^{-1}y^{-12}$

Step 2: Represent the negative exponents as positive ones by moving them to the denominator:

${\color{Red} \frac{x^{10}}{zy^{12}}}$

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Question

Simplify

$\frac{x^5y^6}{x^3y^{10}}$

Answer

When dividing variables with exponents, you subtract the denominator exponents from the numerator exponents of the corresponding variable:

$x^{5-3}y^{6-10}$ , then simplify: $x^2y^{-4}$ .

In order to remove the negative exponent, we move it to the denominator.

Therefore, the answer is $\frac{x^2}{y^4}$ .

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Question

Simplify $x^3\cdot x^{-2}\cdot x^4$ .

Answer

When multiplying variables with the same base, you add the exponents. So, $x^{3+(-2)+4}=x^{1+4}=x^5$ .

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