Negative Indices 3
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Question 1 of 5
1. Question
Simplify`[(-a^2)/(b^3)]^(-5)`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$Negative Powers
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$A negative power means we flip a fraction and make the power positive.`[(-a^2)/(b^3)]^(-5)` `=` $$\left[\frac{b^{\color{#007DDC}{3}}}{-a^{\color{#007DDC}{3}}}\right]^\color{#9a00c7}{5}$$ `=` $$\frac{b^{\color{#007DDC}{3} \times \color{#9a00c7}{5}}}{-a^{\color{#007DDC}{2} \times \color{#9a00c7}{5}}}$$ `=` `(b^(15))/(-a^(10))` `(b^(15))/(-a^(10))` -
Question 2 of 5
2. Question
Simplify`(m^(-3) n^5)/(z^(-4))`Hint
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Negative Powers
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$$$\frac{1}{a^{-\color{#e65021}{n}}}=a^\color{#e65021}{n}$$First, separate the bases..`(m^(-3) n^5)/(z^(-4))` `=` `(m^(-3))/1 xx (n^5)/1 xx 1/(z^(-4))` Now use Negative Powers to simplify the expression.$$\frac{m^{\color{#e65021}{-3}}}{1} \times \frac{n^5}{1} \times \frac{1}{z^{\color{#e65021}{-4}}}$$ `=` $$\frac{1}{m^{\color{#e65021}{3}}} \times \frac{n^5}{1} \times \frac{z^{\color{#e65021}{4}}}{1}$$ `=` `(n^5 z^4)/(m^3)` `(n^5 z^4)/(m^3)` -
Question 3 of 5
3. Question
Simplify`((10x^2 y^(-2))^3)/((2x^(-1)y^3)^2)`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$Negative Powers
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$Use Power of a Power and Negative Powers to simplify the expression.$$\frac{(10x^{\color{#007DDC}{2}}y^{\color{#007DDC}{-2}})^{\color{#9a00c7}{3}}}{(2x^{\color{#007DDC}{-1}}y^{\color{#007DDC}{3}})^{\color{#9a00c7}{2}}}$$ `=` $$\frac{10^{\color{#9a00c7}{3}}x^{\color{#007DDC}{2} \times \color{#9a00c7}{3}}y^{\color{#007DDC}{-2} \times \color{#9a00c7}{3}}}{2^{\color{#9a00c7}{2}}x^{\color{#007DDC}{-1} \times \color{#9a00c7}{2}}y^{\color{#007DDC}{3} \times \color{#9a00c7}{2}}}$$ `=` $$\frac{1000x^6 y^{\color{#e65021}{-6}}}{4x^{\color{#e65021}{-2}}y^6}$$ `=` $$\frac{1000x^6 \times x^{\color{#e65021}{2}}}{4y^6 \times y^{\color{#e65021}{6}}}$$ Finally, use the Product of Powers to simplify the expression further.$$\frac{1000\color{#00880A}{x}^6 \times \color{#00880A}{x}^2}{4y^6 \times y^6}$$ `=` $$\frac{1000\color{#00880A}{x}^{6+2}}{4y^6 \times y^6}$$ `=` $$\frac{1000x^8}{4\color{#00880A}{y}^6 \times \color{#00880A}{y}^6}$$ `=` $$\frac{1000x^8}{4\color{#00880A}{y}^{6+6}}$$ `=` $$\frac{250x^8}{y^{12}}$$ `(1000)/4=250` `(250x^8)/(y^(12))` -
Question 4 of 5
4. Question
Simplify`(-3m^(-2))/(-4m^(-3))`Hint
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Negative Powers
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$Quotient of Powers
$${\color{#00880A}{a}^m}\div{\color{#00880A}{a}^n}=\frac{{\color{#00880A}{a}^m}}{{\color{#00880A}{a}^n}}=\color{#00880A}{a}^{m-n}$$First, separate the bases.`(-3m^(-2))/(-4m^(-3))` `=` `((-3)/(-4))((m^(-2))/(m^(-3)))` `=` `(3/4)((m^(-2))/(m^(-3)))` Recall, 
Now, use the Quotient of Powers to simplify the expression further.$$\left(\frac{3}{4}\right)\left(\frac{\color{#00880A}{m}^{-2}}{\color{#00880A}{m}^{-3}}\right)$$ `=` $$\left(\frac{3}{4}\right)\left(\color{#00880A}{m}^{-2-(-3)}\right)$$ `=` $$\left(\frac{3}{4}\right)\left(\color{#00880A}{m}^{-2+3}\right)$$ `=` `3/4m` A power of `1` does not need to be written `3/4m` -
Question 5 of 5
5. Question
Simplify`[(35x^(-2) y^4)/(7x^4 y^(-6))]^(-1)`Hint
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Negative Powers
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$Quotient of Powers
$${\color{#00880A}{a}^m}\div{\color{#00880A}{a}^n}=\frac{{\color{#00880A}{a}^m}}{{\color{#00880A}{a}^n}}=\color{#00880A}{a}^{m-n}$$A negative power means we flip a fraction and make the power positive.`[(35x^(-2) y^4)/(7x^4 y^(-6))]^(-1)` `=` `[(7x^4 y^(-6))/(35x^(-2) y^4)]^1` `=` `(7x^4 y^(-6))/(35x^(-2) y^4)` A power of `1` does not need to be written `=` $$\frac{7x^4 y^{-6} \color{#CC0000}{\div7}}{35x^{-2} y^4 \color{#CC0000}{\div7}}$$ Divide the numerator and denominator by `7` `=` $$\frac{x^4 y^{-6}}{5x^{-2} y^4}$$ Now, use the Quotient of Powers to simplify the expression further.$$\frac{\color{#00880A}{x}^4 y^{-6}}{5\color{#00880A}{x}^{-2} y^4}$$ `=` $$\frac{\color{#00880A}{x}^{4-(-2)}y^{-6}}{5y^4}$$ `=` $$\frac{\color{#00880A}{x}^{4+2}y^{-6}}{5y^4}$$ `=` $$\frac{x^6\color{#00880A}{y}^{-6}}{5\color{#00880A}{y}^4}$$ `=` $$\frac{x^6\color{#00880A}{y}^{-6-4}}{5}$$ `=` $$\frac{x^6\color{#00880A}{y}^{-6+(-4)}}{5}$$ `=` $$\frac{x^6 y^{\color{#e65021}{-10}}}{5}$$ `=` $$\frac{x^6}{5y^{\color{#e65021}{10}}}$$ `(x^6)/(5y^(10))`
Quizzes
- Index Notation 1
- Index Notation 2
- Index Notation 3
- Multiply Indices 1
- Multiply Indices 2
- Multiply Indices 3
- Multiply Indices 4
- Divide Indices 1
- Divide Indices 2
- Powers of a Power 1
- Powers of a Power 2
- Powers of a Power 3
- Powers of a Power 4
- Zero Powers 1
- Zero Powers 2
- Negative Indices 1
- Negative Indices 2
- Negative Indices 3
- Fractional Indices 1
- Fractional Indices 2
- Fractional Indices 3
- Mixed Operations with Indices 1
- Mixed Operations with Indices 2