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Answer :
Sure, let's simplify each of these expressions step-by-step:
4.3.1 Simplify [tex]\( a^6 \times a^2 \)[/tex]:
When you're multiplying expressions with the same base, you add the exponents. So:
[tex]\[ a^6 \times a^2 = a^{6+2} = a^8 \][/tex]
4.3.2 Simplify [tex]\( z^4 \div z^6 \)[/tex]:
When dividing expressions with the same base, you subtract the exponents. So:
[tex]\[ z^4 \div z^6 = z^{4-6} = z^{-2} \][/tex]
4.3.5 Simplify [tex]\( \sqrt[3]{127-2} \times (-5)^2 \)[/tex]:
First, calculate [tex]\( 127 - 2 = 125 \)[/tex].
Next, find the cube root of 125, which is 5, because [tex]\( 5^3 = 125 \)[/tex].
Then, calculate [tex]\( (-5)^2 = 25 \)[/tex].
Now multiply these results together:
[tex]\[ \sqrt[3]{125} \times 25 = 5 \times 25 = 125 \][/tex]
4.3.6 Simplify [tex]\( \frac{\sqrt{81}}{3}+\frac{\sqrt[3]{216}}{3} \)[/tex]:
First, find [tex]\( \sqrt{81} \)[/tex]. The square root of 81 is 9, because [tex]\( 9^2 = 81 \)[/tex].
Divide this result by 3:
[tex]\[ \frac{9}{3} = 3 \][/tex]
Next, find [tex]\( \sqrt[3]{216} \)[/tex]. The cube root of 216 is 6, because [tex]\( 6^3 = 216 \)[/tex].
Divide this result by 3:
[tex]\[ \frac{6}{3} = 2 \][/tex]
Add these two results together:
[tex]\[ 3 + 2 = 5 \][/tex]
These are the simplified results for each part of the problem.
4.3.1 Simplify [tex]\( a^6 \times a^2 \)[/tex]:
When you're multiplying expressions with the same base, you add the exponents. So:
[tex]\[ a^6 \times a^2 = a^{6+2} = a^8 \][/tex]
4.3.2 Simplify [tex]\( z^4 \div z^6 \)[/tex]:
When dividing expressions with the same base, you subtract the exponents. So:
[tex]\[ z^4 \div z^6 = z^{4-6} = z^{-2} \][/tex]
4.3.5 Simplify [tex]\( \sqrt[3]{127-2} \times (-5)^2 \)[/tex]:
First, calculate [tex]\( 127 - 2 = 125 \)[/tex].
Next, find the cube root of 125, which is 5, because [tex]\( 5^3 = 125 \)[/tex].
Then, calculate [tex]\( (-5)^2 = 25 \)[/tex].
Now multiply these results together:
[tex]\[ \sqrt[3]{125} \times 25 = 5 \times 25 = 125 \][/tex]
4.3.6 Simplify [tex]\( \frac{\sqrt{81}}{3}+\frac{\sqrt[3]{216}}{3} \)[/tex]:
First, find [tex]\( \sqrt{81} \)[/tex]. The square root of 81 is 9, because [tex]\( 9^2 = 81 \)[/tex].
Divide this result by 3:
[tex]\[ \frac{9}{3} = 3 \][/tex]
Next, find [tex]\( \sqrt[3]{216} \)[/tex]. The cube root of 216 is 6, because [tex]\( 6^3 = 216 \)[/tex].
Divide this result by 3:
[tex]\[ \frac{6}{3} = 2 \][/tex]
Add these two results together:
[tex]\[ 3 + 2 = 5 \][/tex]
These are the simplified results for each part of the problem.
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