Write $2 \times 2 \times 2 \times 2 \times 2 \times 2$ in the form $2^x$ where $x \in \mathbb{N}$ - Junior Cycle Mathematics - Question 7 - 2014
Question 7
Write $2 \times 2 \times 2 \times 2 \times 2 \times 2$ in the form $2^x$ where $x \in \mathbb{N}$.
If $a^p \times a^3 = a^8$, write down the value of $p$.
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Worked Solution & Example Answer:Write $2 \times 2 \times 2 \times 2 \times 2 \times 2$ in the form $2^x$ where $x \in \mathbb{N}$ - Junior Cycle Mathematics - Question 7 - 2014
Step 1
Write $2 \times 2 \times 2 \times 2 \times 2 \times 2$ in the form $2^x$ where $x \in \mathbb{N}$.
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Answer
To simplify 2×2×2×2×2×2, we can count the number of factors of 2 in the multiplication. There are 6 factors of 2, so we can write it as:
26.
Step 2
If $a^p \times a^3 = a^8$, write down the value of $p$.
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Answer
Using the laws of exponents, we can combine terms on the left: ap×a3=ap+3.
Setting this equal to a8, we have: ap+3=a8.
Thus, we can equate the exponents: p+3=8.
Solving for p, we find: p=8−3=5.
Therefore, the value of p is 5.
Step 3
Simplify $\frac{2^5 \times 2^6}{2^4 \times 2^3}$.
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Answer
First, we can combine the terms in the numerator:
25×26=25+6=211.
Next, we simplify the denominator:
24×23=24+3=27.
Now we substitute these back into the fraction:
27211=211−7=24.
Thus, the simplified expression is:
24.
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