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Mixed Operations with Exponents 1Mixed Operations with Exponents 1
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Question 1 of 5
1. Question
Simplify`(8b^3)^2 b^5`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$First, apply the power of a power to all terms inside the brackets, then simplify.$${(8b^{\color{#007DDC}{3}})}^{\color{#9a00c7}{2}} b^5$$ `=` $$(8^{\color{#9a00c7}{2}}b^{\color{#007DDC}{3} \times \color{#9a00c7}{2}}) b^{5}$$ `=` `64b^6 b^5` Simplify further by applying the Product of Powers to the values with the same base.$$64\color{#00880A}{b}^6 \color{#00880A}{b}^5$$ `=` $$64\color{#00880A}{b}^{6+5}$$ `=` `64b^(11)` `64b^(11)` -
Question 2 of 5
2. Question
Simplify`(3m^3)^2xx m^4 xx z^0`Hint
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Power of Zero
$$\color{#A57200}{a^0} =\color{#A57200}{1}$$Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$First, apply the Power of Zero to the third term.`(3m^3)^2xx m^4 xx` `z^0` `=` `(3m^3)^2xx m^4 xx` `1` `=` `(3m^3)^2xx m^4` Next, apply the power of a power to the first term.$${(3m^{\color{#007DDC}{3}})}^{\color{#9a00c7}{2}} \times m^4$$ `=` $$(3^{\color{#9a00c7}{2}}m^{\color{#007DDC}{3} \times \color{#9a00c7}{2}}) \times m^{4}$$ `=` `9m^6 xx m^4` Simplify further by applying the Product of Powers to the values with the same base.$$9\color{#00880A}{m}^6 \times \color{#00880A}{m}^4$$ `=` $$9\color{#00880A}{m}^{6+4}$$ `=` `9m^10` `9m^10` -
Question 3 of 5
3. Question
Simplify`(-5a^2 b)^3 (4a^3 b^5)^2`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$Apply the power of a power to the first bracket.$${(-5a^{\color{#007DDC}{2}} b)}^{\color{#9a00c7}{3}} {(4a^3 b^5)}^2$$ `=` $$(-5^{\color{#9a00c7}{3}}a^{\color{#007DDC}{2} \times \color{#9a00c7}{3}} b^{\color{#9a00c7}{3}}) {(4a^3 b^5)}^2$$ `=` `(-125a^6 b^3) (4a^3 b^5)^2` Do the same to the second bracket.$$(-125a^6 b^3) {(4a^\color{#007DDC}{3} b^\color{#007DDC}{5})}^\color{#9a00c7}{2}$$ `=` $$(-125a^6 b^3) (4^{\color{#9a00c7}{2}}a^{\color{#007DDC}{3} \times \color{#9a00c7}{2}} b^{\color{#007DDC}{5} \times \color{#9a00c7}{2}})$$ `=` `(-125a^6 b^3) (16a^6 b^10)` Simplify further by applying the Product of Powers to the values with the same base.$$(-125\color{#00880A}{a}^6 b^3) (16\color{#00880A}{a}^6 b^{10})$$ `=` $$- 125 \times 16 \times \color{#00880A}{a}^{6+6} \times b^3 \times b^{10}$$ Collect like terms `=` $$-2000 \times a^{12} \times \color{#00880A}{b}^3 \times \color{#00880A}{b}^{10}$$ Evaluate the constant `=` $$-2000 \times a^{12} \times \color{#00880A}{b}^{3+10}$$ `=` $$-2000 \times a^{12} \times b^{13}$$ `=` `-2000a^12 b^13` `-2000a^12 b^13` -
Question 4 of 5
4. Question
Simplify`(3x^3 y^(-2))/(-6xy^(-5))`Hint
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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, collect like terms.`(3x^3 y^(-2))/(-6xy^(-5))` `=` `(3/(-6))((x^3)/(x))((y^(-2))/(y^(-5)))` Use Quotient of Powers to simplify the fractions.$$\left(\frac{3}{-6}\right) \left(\frac{\color{#00880A}{x}^3}{\color{#00880A}{x}}\right) \left(\frac{y^{-2}}{y^{-5}}\right)$$ `=` $$-\frac{1}{2} \left(\color{#00880A}{x}^{3-1}\right) \left(\frac{y^{-2}}{y^{-5}}\right)$$ Simplify the fraction `=` $$-\frac{1}{2} x^2 \left(\frac{\color{#00880A}{y}^{-2}}{\color{#00880A}{y}^{-5}}\right)$$ `=` $$-\frac{1}{2} x^2 \color{#00880A}{y}^{-2-(-5)}$$ `=` `-1/2 x^2 y^3` `-1/2 x^2 y^3` -
Question 5 of 5
5. Question
Simplify`((x^2)/y)^5 xx 1/(2x^2)`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$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, apply the power of 5 to the top and bottom of the fraction.`((x^2)/y)^5 xx 1/(2x^2)` `=` `((x^2)^5)/(y^5) xx 1/(2x^2)` Next, apply the power of a power to the numerator of the first term.$$\frac{{(x^\color{#007DDC}{2})}^\color{#9a00c7}{5}}{y^5} \times \frac{1}{2x^2}$$ `=` $$\frac{x^{\color{#007DDC}{2} \times \color{#9a00c7}{5}}}{y^5} \times \frac{1}{2x^2}$$ `=` `(x^10)/(y^5) xx 1/(2x^2)` Bring `x` terms together in one fraction.`(x^10)/(y^5) xx 1/(2x^2)` `=` `(x^10)/(x^2) xx 1/(2y^5)` Simplify further by applying the Quotient of Powers to the values with the same base.$$\frac{\color{#00880A}{x}^{10}}{\color{#00880A}{x}^2} \times \frac{1}{2y^5}$$ `=` $$\frac{\color{#00880A}{x}^{10-2}}{1} \times \frac{1}{2y^5}$$ `=` `(x^8)/(2y^5)` `(x^8)/(2y^5)`
Quizzes
- Exponent Notation 1
- Exponent Notation 2
- Exponent Notation 3
- Multiply Exponents (Product Rule) 1
- Multiply Exponents (Product Rule) 2
- Multiply Exponents (Product Rule) 3
- Multiply Exponents (Product Rule) 4
- Divide Exponents (Quotient Rule) 1
- Divide Exponents (Quotient Rule) 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 Exponents 1
- Negative Exponents 2
- Negative Exponents 3
- Rational Exponents 1
- Rational Exponents 2
- Rational Exponents 3
- Mixed Operations with Exponents 1
- Mixed Operations with Exponents 2