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Write each of the following in the form $2^n$, where $n \in \mathbb{Q}$ - Junior Cycle Mathematics - Question 10 - 2016

Question 10

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Write each of the following in the form $2^n$, where $n \in \mathbb{Q}$. (a) $2^3 \times 2^5 \times 2^{10}$ (b) $8^{25}$ (c) $\sqrt{8}$

Worked Solution & Example Answer:Write each of the following in the form $2^n$, where $n \in \mathbb{Q}$ - Junior Cycle Mathematics - Question 10 - 2016

Step 1

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Answer

To simplify the expression, we use the property of exponents that states $a^m \times a^n = a^{m+n}$ . Thus, we have:

2^3 \times 2^5 \times 2^{10} = 2^{3+5+10} = 2^{18}.

Therefore, the final answer is $2^{18}$ .

Step 2

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Answer

We can express 8 as a power of 2: $8 = 2^3$ . Hence, we rewrite $8^{25}$ :

8^{25} = (2^3)^{25}.

Applying the power of a power rule, $(a^m)^n = a^{m \times n}$ , we get:

(2^3)^{25} = 2^{3 \times 25} = 2^{75}.

Thus the answer is $2^{75}$ .

Step 3

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Answer

To simplify $\sqrt{8}$ , we recall that $\sqrt{a} = a^{1/2}$ . We can express 8 as $8 = 2^3$ , so:

\sqrt{8} = \sqrt{(2^3)} = (2^3)^{1/2}.

Utilizing the power of a power rule:

(2^3)^{1/2} = 2^{3 \times \frac{1}{2}} = 2^{\frac{3}{2}}.

Therefore, the answer is $2^{\frac{3}{2}}$ .

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