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Mixed Operations with Exponents 2Mixed Operations with Exponents 2
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Question 1 of 5
1. Question
Simplify`(2x^(2/3))^3xx(9x^4)^(1/2)`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$Fractional Powers
$$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$$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.$$(2x^{\color{#007DDC}{\frac{2}{3}}})^{\color{#9a00c7}{3}} \times (9x^{\color{#007DDC}{4}})^{\color{#9a00c7}{\frac{1}{2}}}$$ `=` $$(2^{\color{#9a00c7}{3}}\times x^{\color{#007DDC}{\frac{2}{3}} \times \color{#9a00c7}{3}})\times(9^{\color{#9a00c7}{\frac{1}{2}}}\times x^{\color{#007DDC}{4} \times \color{#9a00c7}{\frac{1}{2}}})$$ `=` `8x^2xx9^(1/2)xx x^2` `2/3xx3=2` and `4xx1/2=2` Next, use Fractional Powers to simplify the expression further.$$8x^2 \times 9^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{2}}} \times x^2$$ `=` $$8x^2 \times (\sqrt[\color{#D800AD}{2}]{9})^{\color{#004ec4}{1}} \times x^2$$ `=` `8x^2xx(3)^1xxx^2` `root (2)(9)=3` `=` `8x^2xxx^2xx3` Rearrange the values Finally, simplify further by applying the Product of Powers to the values with the same base.$$8\color{#00880A}{x}^2\times \color{#00880A}{x}^2 \times 3$$ `=` $$8 \times 3 \times \color{#00880A}{x}^{2+2}$$ `=` `24x^4` `24x^4` -
Question 2 of 5
2. Question
Simplify`((x^4)/(y^3))^3 xx 4/(7x^5)`Hint
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Power of a Power
$${(a^\color{#007DDC}{m})}^{\color{#9a00c7}{n}}=a^{\color{#007DDC}{m} \times \color{#9a00c7}{n}}$$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 3 to the top and bottom of the fraction.`((x^4)/(y^3))^3 xx 4/(7x^5)` `=` `((x^4)^3)/((y^3)^3) xx 4/(7x^5)` Next, apply the power of a power to the first term.$$\frac{{(x^\color{#007DDC}{4})}^\color{#9a00c7}{3}}{{(y^\color{#007DDC}{3})}^\color{#9a00c7}{3}} \times \frac{4}{7x^5}$$ `=` $$\frac{x^{\color{#007DDC}{4} \times \color{#9a00c7}{3}}}{y^{\color{#007DDC}{3} \times \color{#9a00c7}{3}}} \times \frac{4}{7x^5}$$ `=` `(x^12)/(y^9) xx 4/(7x^5)` Bring `x` terms together in one fraction.`(x^12)/(y^9) xx 4/(7x^5)` `=` `(x^12)/(x^5) xx 4/(7y^9)` Simplify further by applying the Quotient of Powers to the values with the same base.$$\frac{\color{#00880A}{x}^{12}}{\color{#00880A}{x}^5} \times \frac{4}{7y^9}$$ `=` $$\frac{\color{#00880A}{x}^{12-5}}{1} \times \frac{4}{7y^9}$$ `=` `(4x^7)/(7y^9)` `(4x^7)/(7y^9)` -
Question 3 of 5
3. Question
Simplify`(4x^(1/4))^2 -: (64x^3)^(1/3)`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
$${\color{#00880A}{a}^m}\div{\color{#00880A}{a}^n}=\frac{{\color{#00880A}{a}^m}}{{\color{#00880A}{a}^n}}=\color{#00880A}{a}^{m-n}$$Negative Powers
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$First, apply the power of a power to all terms inside the brackets, then simplify.$${(4x^{\color{#007DDC}{\frac{1}{4}}})}^\color{#9a00c7}{2} \div {(64x^\color{#007DDC}{3})}^{\color{#9a00c7}{\frac{1}{3}}}$$ `=` $$\left(4^\color{#9a00c7}{2} \times x^{\color{#007DDC}{\frac{1}{4}} \times \color{#9a00c7}{2}} \right) \div \left(64^{\color{#9a00c7}{\frac{1}{3}}} \times x^{\color{#007DDC}{3} \times \color{#9a00c7}{\frac{1}{3}}} \right)$$ `=` `(16 \times x^(1/2)) -: (4 \times x^1)` `1/4xx2=1/2` and `3xx1/3=1` `=` `(16x^(1/2))/(4x^1)` `=` `(4x^(1/2))/(x^1)` `16-:4=4` Simplify further by applying the Quotient of Powers.$$\frac{4\color{#00880A}{x}^\frac{1}{2}}{\color{#00880A}{x}^1}$$ `=` $$4\color{#00880A}{x}^{\frac{1}{2}-1}$$ `=` `4x^(-1/2)` Finally, simplify further by applying Negative Powers.$$4x^{-\color{#e65021}{\frac{1}{2}}}$$ `=` $$\frac{4}{x^\color{#e65021}{\frac{1}{2}}}$$ `=` `4/(sqrtx)` `4/(sqrtx)` -
Question 4 of 5
4. Question
Simplify`(a^2)^(-3/2) -: (b^(-1/2))^3`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
$$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$$Fractional Powers
$$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$$First, apply the power of a power to all terms inside the brackets, then simplify.$${(a^\color{#007DDC}{2})}^\color{#9a00c7}{-\frac{3}{2}} \div {(b^\color{#007DDC}{-\frac{1}{2}})}^\color{#9a00c7}{3}$$ `=` $$\left(a^{\color{#007DDC}{2} \times \left(\color{#9a00c7}{-\frac{3}{2}}\right)}\right) \div \left(b^{\color{#007DDC}{-\frac{1}{2}} \times \color{#9a00c7}{3}}\right)$$ `=` `a^(-3) -: b^(-3/2)` Simplify further by applying Negative Powers.$$a^{-\color{#e65021}{3}} \div b^{-\color{#e65021}{\frac{3}{2}}}$$ `=` $$\frac{1}{a^\color{#e65021}{3}} \div \frac{1}{b^\color{#e65021}{\frac{3}{2}}}$$ Finally, apply Fractional Powers and simplify.$$\frac{1}{a^3} \div \frac{1}{b^\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}}$$ `=` $$\frac{1}{a^3} \div \frac{1}{\left(\sqrt[\color{#D800AD}{2}]{b}\right)^\color{#004ec4}{3}}$$ `=` $$\frac{1}{a^3} \div \frac{1}{\sqrt{b^3}}$$ `=` $$\frac{1}{a^3} \times \frac{\sqrt{b^3}}{1}$$ `=` `(sqrt(b^3))/(a^3)` `(sqrt(b^3))/(a^3)` -
Question 5 of 5
5. Question
Simplify`(1/3 ab^3)^2 [(-3b)^2]^3`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 each term in the brackets.$${\left(\frac{1}{3} ab^\color{#007DDC}{3} \right)}^\color{#9a00c7}{2} \left[{(-3b)}^\color{#007DDC}{2}\right]^\color{#9a00c7}{3}$$ `=` $$\left(\frac{1}{3}\right)^\color{#9a00c7}{2} \times a^\color{#9a00c7}{2} \times b^{\color{#007DDC}{3} \times \color{#9a00c7}{2}} \times {(-3b)}^{\color{#007DDC}{2} \times \color{#9a00c7}{3}}$$ `=` `1/9 a^2 b^6 xx (-3b)^6` `=` `1/9 a^2 b^6 xx (-3)^6 b^6` `=` `((-3)^6)/9 a^2 b^6 b^6` `=` `729/9 a^2 b^6 b^6` `=` `81a^2 b^6 b^6` Simplify further by applying the Product of Powers to the values with the same base.$$81a^2 \color{#00880A}{b}^6 \color{#00880A}{b}^6$$ `=` $$81a^2 \color{#00880A}{b}^{6+6}$$ `=` `81a^2 b^12` `81a^2 b^12`
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