Multiplying and Dividing Exponents - Algebra II

Card 0 of 748

Question

Simplify $x^5y^2x^{-4}$

Answer

First, combine exponents of like variables. This gives us

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Question

Simplify $z^3y^2x^6y^{-3}$

Answer

First, combine exponents of like variables. This gives us $z^3y^{-1}x^6$ which simplifies to $\frac{z^3x^6}{y}$

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Question

Simplify.

$3^{-14}3^{23}3^{-15}$

Answer

When multiplying exponents with the same base, you just have to add the exponents.

$3^{-14}3^{23}3^{-15}=3^{-14+23-15}=3^{-6}$

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Question

Simplify.

$\frac{a^{45}}{a^{30}}$

Answer

When dividing exponents with the same base, we just subtract the exponents.

$\frac{a^{45}}{a^{30}}=a^{45-30}=a^{15}$

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Question

Simplify.

$\frac{a^7}{a^{18}}$

Answer

When dividing exponents with the same base, we just subtract the exponents.

$\frac{a^{7}}{a^{18}}=a^{7-18}=a^{-11}$

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Question

Simplify: $12^{19}\div12^7$

Answer

When dividing exponents with the same base, we subtract the exponents and keep the base the same.

$12^{19}\div12^7=12^{19-7}=12^{12}$

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Question

Simplify:

Answer

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$23^7*23^8=23^{7+8}=23^{15}$

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Question

Simplify: $2^4*2^{-12}$

Answer

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$2^4*2^{-12}=2^{4+(-12)}=2^{-8}$

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Question

Simplify: $(-3)^7*(-3)^{-2}$

Answer

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$(-3)^7*(-3)^{-2}=(-3)^{7+(-2)}=(-3)^{5}$

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Question

Simplify:

Answer

When multiplying exponents with the same base, we just keep the base the same and add the exponents.

$x^5*x^6=x^{5+6}=x^{11}$

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Question

Simplify: $x^{12}*x^{-8}$

Answer

When multiplying exponents with the same base, we just keep the base the same and add the exponents.

$x^{12}*x^{-8}=x^{12+(-8)}=x^4$

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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

Evaluate:

$12^{-4}*12^{5}$

Answer

When multiplying with same base but different exponents, you just add the exponents and keep the base the same.

$12^{-4}*12^{5}=12^{-4+5}=12^{1}=12$

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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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