Rational Exponents 1

Topics

Algebra 1

Exponents

Rational Exponents (Fractional)

Rational Exponents 1

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Question 1 of 6

1. Question
Simplify

$a^{\frac{T}{B}}$ $a^(T/B)$
- 1. $\sqrt[B]{a^{T}}$ $root(B)(a^T)$
- 2. $\sqrt[T]{a^{B}}$ $root(T)(a^B)$
- 3. $B \sqrt{a^{T}}$ $B sqrt(a^T)$
- 4. $T \sqrt{a^{B}}$ $T sqrt(a^B)$
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Simplify

$64^{\frac{2}{3}}$ $64^(2/3)$
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Identify the known lengths

$Base = a$ $\text(Base)=a$

$Numerator/Top = T$ $\text(Numerator/Top)=T$

$Denominator/Bottom = B$ $\text(Denominator/Bottom)=B$

To simplify a fractional power, use the denominator/bottom value $B$ $B$ as a root and use the numerator/top value $T$ $T$ as the power.

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}$ $=$ $=$ $\sqrt[\color{#D800AD}{B}]{a^{\color{#004ec4}{T}}}$

To remember this easily, take note that $B$ $B$ comes before $T$ $T$ alphabetically. This means that $B$ $B$ will be at the left side(the root) and $T$ $T$ will be at the right side (the power).

$\sqrt[B]{a^{T}}$ $root(B)(a^T)$ can also be written as ${(\sqrt[B]{a})}^{T}$ $(root(B)(a))^T$

$\sqrt[B]{a^{T}}$ $root(B)(a^T)$

Question 2 of 6

Simplify

64^{\frac{2}{3}}

$64^(2/3)$

(16)

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Fractional Powers

a^{\frac{T}{B}} = (\sqrt[B]{a})^{T}

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

Use Fractional Powers to simplify the expression.

$64^{\frac{\color{#004ec4}{2}}{\color{#D800AD}{3}}}$	$=$ $=$	$(\sqrt[\color{#D800AD}{3}]{64})^{\color{#004ec4}{2}}$
	$=$ $=$	${(4)}^{2}$ $(4)^2$	$\sqrt[3]{64} = 4$ $root (3)(64)=4$
	$=$ $=$	$16$ $16$

16

$16$

Question 3 of 6

3. Question
Simplify

$27^{\frac{1}{3}}$ $27^(1/3)$
- (3)
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Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

Use Fractional Powers to simplify the expression.

$27^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{3}}}$ $=$ $=$ $(\sqrt[\color{#D800AD}{3}]{27})^{\color{#004ec4}{1}}$

$=$ $=$ ${(3)}^{1}$ $(3)^1$ $\sqrt[3]{27} = 3$ $root (3)(27)=3$

$=$ $=$ $3$ $3$

$3$ $3$
Question 4 of 6

4. Question
Simplify

$9^{\frac{1}{2}}$ $9^(1/2)$
- (3)
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Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

Use Fractional Powers to simplify the expression.

$9^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{2}}}$ $=$ $(\sqrt[\color{#D800AD}{2}]{9})^{\color{#004ec4}{1}}$

$=$ $(3)^1$ $root (2)(9)=3$

$=$ $3$

$3$
Question 5 of 6

5. Question
Simplify

$25^(3/2)$
- (125)
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Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

Use Fractional Powers to simplify the expression.

$25^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}}$ $=$ $(\sqrt[\color{#D800AD}{2}]{25})^{\color{#004ec4}{3}}$

$=$ $(5)^3$ $root (2)(25)=5$

$=$ $125$

$125$
Question 6 of 6

6. Question
Simplify

$9^(3/2)$
- (27)
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Fractional Powers

$a^{\frac{\color{#004ec4}{T}}{\color{#D800AD}{B}}}=(\sqrt[\color{#D800AD}{B}]{a})^{\color{#004ec4}{T}}$

Use Fractional Powers to simplify the expression.

$9^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}}$ $=$ $(\sqrt[\color{#D800AD}{2}]{9})^{\color{#004ec4}{3}}$

$=$ $(3)^3$ $root (2)(9)=3$

$=$ $27$

$27$

Rational Exponents 1

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Quiz summary

Information

1. Question

Hint

Identify the known lengths

2. Question

Hint

Keep Going!

Fractional Powers

3. Question

Hint

Correct!

Fractional Powers

4. Question

Hint

Keep Going!

Fractional Powers

5. Question

Hint

Fantastic!

Fractional Powers

6. Question

Hint

Excellent!

Fractional Powers

Quizzes

$27^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{3}}}$	$=$ $=$	$(\sqrt[\color{#D800AD}{3}]{27})^{\color{#004ec4}{1}}$
	$=$ $=$	${(3)}^{1}$ $(3)^1$	$\sqrt[3]{27} = 3$ $root (3)(27)=3$
	$=$ $=$	$3$ $3$

$9^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{2}}}$	$=$	$(\sqrt[\color{#D800AD}{2}]{9})^{\color{#004ec4}{1}}$
	$=$	$(3)^1$	$root (2)(9)=3$
	$=$	$3$

$25^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}}$	$=$	$(\sqrt[\color{#D800AD}{2}]{25})^{\color{#004ec4}{3}}$
	$=$	$(5)^3$	$root (2)(25)=5$
	$=$	$125$

$9^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}}$	$=$	$(\sqrt[\color{#D800AD}{2}]{9})^{\color{#004ec4}{3}}$
	$=$	$(3)^3$	$root (2)(9)=3$
	$=$	$27$