The distributive property of exponents over multiplication says that when you have multiple factors raised to an exponent, the exponent can be distributed to each of the factors individually.
\((ab)^{x}=a^{x}b^{x}\)
This property can also be applied to division and it can be used in reverse to multiply or divide powers with the same exponent.
Most math teachers call the distributive property of exponents some variation of...
I call it the "distributive property of exponents over multiplication" because the relationship between exponents and multiplication is very similar to the relationship between multiplication and addition.
The relationship between multiplication and addition is called the "distributive property of multiplication over addition" and you can read more about it here.
\(3(2x+5y)=6x+15y\)
\((x^{5}y^{4})^{2}=x^{10}y^{8}\)
There are two different distributive properties.
There are a lot of similarities between these two properties.
Both distributive properties have parentheses.
When you look at WHY they work, there are also a lot of similarities...
\(3(2x+5y)=6x+15y\)
\(2x+5y\) is the same thing as \(x+x+y+y+y+y+y\).
When you make 3 copies of that, you get...
\(x+x+y+y+y+y+y+\)
\(x+x+y+y+y+y+y+\)
\(x+x+y+y+y+y+y\)
There are a total of 6 x's and 15 y's being added together, so that is why...
\(3(2x+5y)=6x+15y\)
\((x^{5}y^{4})^{2}=x^{10}y^{8}\)
\(x^{5}y^{4}\) is the same thing as \(xxxxxyyyy\).
When you multiply that by itself 2 times, you get...
\((xxxxxyyyy)(xxxxxyyyy)\)
There's a total of 10 x's and 8 y's multiplied together, so that is why...
\((x^{5}y^{4})^{2}=x^{10}y^{8}\)
Exponents can NOT be distributed over addition or subtraction.
\((x+y)^2\neq x^2+y^2\)
If you have an exponent outside of parentheses with addition or subtraction inside, then you need to follow the rules for multiplying polynomials.
\((ab)^{x}=a^{x}b^{x}\)
\((3x)^{4}=81x^{4}\)
\((ab)^{5}=a^{5}b^{5}\)
\((2y)^{5}=32y^{5}\)
Simplify \((3x^{4}y)^2\)
The factors in this expression are 3, \(x^4\), and y.
I need to distribute the outside exponent of 2 to each factor.
\((3x^{4}y)^2=3^{2}(x^4)^{2}y^{2}\)
Next, I'll simplify each factor, if possible.
\(3^{2} = 3\times3=9\)
\((x^4)^{2}=x^8\) because of the power rule of exponents.
\(y^2\) cannot be simplified further.
\(3^{2}(x^4)^{2}y^{2}=9x^{8}y^{2}\)
Answer: \((3x^{4}y)^2=9x^{8}y^{2}\)
\((\frac{a}{b})^{x}=\frac{a^{x}}{b^{x}}\)
\((\frac{a}{b})^{7}=\frac{a^{7}}{b^{7}}\)
\((\frac{x^2}{4})^{3}=\frac{x^{6}}{64}\)
\((\frac{5}{xyz})^{3}=\frac{125}{x^{3}y^{3}z^{3}}\)
Simplify \((\frac{3xy}{7z^4})^{2}\)
The factors in the numerator are 3, x, and y.
The factors in the denominator are 7 and \(z^4\).
I need to distribute the outside exponent of 2 to each factor.
\((\frac{3xy}{7z^4})^{2}=\frac{3^2x^2y^2}{7^2(z^4)^2}\)
Next, I'll simplify each factor, if possible.
\(3^{2} = 3\times3=9\)
\(x^2\) and \(y^2\) cannot be simplified further.
\(7^{2} = 7\times7=49\)
\((x^4)^{2}=x^8\) because of the power rule of exponents.
So...
\(\frac{3^2x^2y^2}{7^2(z^4)^2}=\frac{9x^2y^2}{49z^8}\)
Answer: \((\frac{3xy}{7z^4})^{2}=\frac{9x^2y^2}{49z^8}\)
\(a^{x}b^{x}=(ab)^{x}\)
\(2^{3}4^{3}=8^{3}\)
\(7^{2}x^{2}=(7x)^{2}\)
\(a^{5}b^{5}=(ab)^{5}\)
Simplify \((6^3)(5^3)\).
These two powers do have the same exponent (3).
So, I will multiply the bases together.
\((6^3)(5^3)=30^3\)
There are no variables so I can evaluate this expression.
\(30^{3} = 30\times30\times30=27000\)
Answer: \((6^3)(5^3)=27000\)
\(\frac{a^{x}}{b^{x}}=(\frac{a}{b})^{x}\)
\(\frac{a^{9}}{b^{9}}=(\frac{a}{b})^{9}\)
\(\frac{4^{2}}{x^{2}}=(\frac{4}{x})^{2}\)
\(\frac{5^3x^{3}}{y^{3}}=(\frac{5x}{y})^{3}\)
Simplify \(\frac{8^{3}}{2^{3}}\).
These two powers do have the same exponent (3).
So, I will divide the bases and rewrite the power.
\(\frac{8^{3}}{2^{3}}=4^3\)
There are no variables so I can evaluate the expression.
\(4^{3} = 4\times4\times4=64\)
Answer:\(\frac{8^{3}}{2^{3}}=64\)
Khan Academy - Powers of products & quotients (structured practice)
Khan Academy - Powers of products & quotients (integer exponents)
Khan Academy - Powers of products & quotients
Math Aids - Products to a Power Worksheet Generator
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Math Aids - Products and Quotients to a Power Worksheet Generator
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