Fractional Indices 2

Years

Year 7

Indices

Fractional Indices

Fractional Indices 2

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Question 1 of 6

1. Question
Simplify

$16^{\frac{3}{4}}$ $16^(3/4)$
- (8)
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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.

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

$=$ $=$ ${(2)}^{3}$ $(2)^3$ $\sqrt[4]{16} = 2$ $root (4)(16)=2$

$=$ $=$ $8$ $8$

$8$ $8$

Question 2 of 6

Simplify

125^{\frac{1}{3}}

$125^(1/3)$

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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.

$125^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{3}}}$	$=$ $=$	$(\sqrt[\color{#D800AD}{3}]{125})^{\color{#004ec4}{1}}$
	$=$ $=$	${(5)}^{1}$ $(5)^1$	$\sqrt[3]{125} = 5$ $root (3)(125)=5$
	$=$ $=$	$5$ $5$

5

$5$

Question 3 of 6

Simplify

x \sqrt{x}

$xsqrtx$

1. $x^{\frac{3}{2}}$ $x^(3/2)$
2. $\frac{1}{2 x}$ $1/(2x)$
3. $x^{2}$ $x^2$
4. $2 x$ $2x$

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Fractional Powers

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

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

Product of Powers

a^{m} \times a^{n} = a^{m + n}

${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$

Fractional Powers

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

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

Use Fractional Powers to simplify the expression.

x^{1} \sqrt[2]{27^{1}}

$x^1\sqrt[\color{#D800AD}{2}]{27^{\color{#004ec4}{1}}}$

=

$=$

x^{1} \times x^{\frac{1}{2}}

$x^1 \times x^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{2}}}$

Next, use the Product of Powers to simplify the expression further.

$\color{#00880A}{x}^1 \times \color{#00880A}{x}^{\frac{1}{2}}$	$=$ $=$	$\color{#00880A}{x}^{1+\frac{1}{2}}$
	$=$ $=$	$\color{#00880A}{x}^{1\frac{1}{2}}$
	$=$ $=$	$x^{\frac{3}{2}}$ $x^(3/2)$	Convert the power into a fraction

x^{\frac{3}{2}}

$x^(3/2)$

Question 4 of 6

Simplify

16^{- \frac{1}{2}}

$16^{-\frac{1}{2}}$

Write fractions as “a/b”

(1/4)

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Negative Power

a^{- n} = \frac{1}{a^{n}}

$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$

Fractional Powers

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

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

First, use Negative Powers to simplify the expression.

16^{- \frac{1}{2}}

$16^{-\color{#e65021}{\frac{1}{2}}}$

=

$=$

\frac{1}{16^{\frac{1}{2}}}

$\frac{1}{16^\color{#e65021}{\frac{1}{2}}}$

Next, simplify further using Fractional Powers.

$\frac{1}{16^{\frac{\color{#004ec4}{1}}{\color{#D800AD}{2}}}}$	$=$ $=$	$\frac{1}{(\sqrt[\color{#D800AD}{2}]{16})^{\color{#004ec4}{1}}}$
	$=$ $=$	$\frac{1}{{(4)}^{1}}$ $1/(4)^1$	$\sqrt[2]{16} = 4$ $root (2)(16)=4$
	$=$ $=$	$\frac{1}{4}$ $1/4$	$4^{1} = 4$ $4^1=4$

\frac{1}{4}

$1/4$

Question 5 of 6

Simplify

4^{- \frac{3}{2}}

$4^{-\frac{3}{2}}$

Write fractions as “a/b”

(1/8)

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Negative Power

a^{- n} = \frac{1}{a^{n}}

$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$

Fractional Powers

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

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

First, use Negative Powers to simplify the expression.

4^{- \frac{3}{2}}

$4^{-\color{#e65021}{\frac{3}{2}}}$

=

$=$

\frac{1}{4^{\frac{3}{2}}}

$\frac{1}{4^\color{#e65021}{\frac{3}{2}}}$

Next, simplify further using Fractional Powers.

$\frac{1}{4^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{2}}}}$	$=$ $=$	$\frac{1}{(\sqrt[\color{#D800AD}{2}]{4})^{\color{#004ec4}{3}}}$
	$=$ $=$	$\frac{1}{{(2)}^{3}}$ $1/(2)^3$	$\sqrt[2]{4} = 2$ $root (2)(4)=2$
	$=$ $=$	$\frac{1}{8}$ $1/8$	$2^{3} = 8$ $2^3=8$

\frac{1}{8}

$1/8$

Question 6 of 6

Simplify

8^{- \frac{4}{3}}

$8^{-\frac{4}{3}}$

Write fractions as “a/b”

(1/16)

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Negative Power

a^{- n} = \frac{1}{a^{n}}

$a^{-\color{#e65021}{n}}=\frac{1}{a^\color{#e65021}{n}}$

Fractional Powers

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

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

First, use Negative Powers to simplify the expression.

8^{- \frac{4}{3}}

$8^{-\color{#e65021}{\frac{4}{3}}}$

=

$=$

\frac{1}{8^{\frac{4}{3}}}

$\frac{1}{8^\color{#e65021}{\frac{4}{3}}}$

Next, simplify further using Fractional Powers.

$\frac{1}{8^{\frac{\color{#004ec4}{4}}{\color{#D800AD}{3}}}}$	$=$ $=$	$\frac{1}{(\sqrt[\color{#D800AD}{3}]{8})^{\color{#004ec4}{4}}}$
	$=$ $=$	$\frac{1}{{(2)}^{4}}$ $1/(2)^4$	$\sqrt[3]{8} = 2$ $root (3)(8)=2$
	$=$ $=$	$\frac{1}{16}$ $1/(16)$	$2^{4} = 16$ $2^4=16$

\frac{1}{16}

$1/(16)$

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Quiz summary

Information

1. Question

Hint

Nice Job!

Fractional Powers

2. Question

Hint

Well Done!

Fractional Powers

3. Question

Hint

Fantastic!

Fractional Powers

Product of Powers

Fractional Powers

4. Question

Hint

Nice Job!

Negative Power

Fractional Powers

5. Question

Hint

Excellent!

Negative Power

Fractional Powers

6. Question

Hint

Keep Going!

Negative Power

Fractional Powers

Quizzes

$16^{\frac{\color{#004ec4}{3}}{\color{#D800AD}{4}}}$	$=$ $=$	$(\sqrt[\color{#D800AD}{4}]{16})^{\color{#004ec4}{3}}$
	$=$ $=$	${(2)}^{3}$ $(2)^3$	$\sqrt[4]{16} = 2$ $root (4)(16)=2$
	$=$ $=$	$8$ $8$