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Answer :
Sure! Let's simplify each expression step-by-step:
4.3.1: [tex]\(a^6 \times a^2\)[/tex]
To simplify this, we use the property of exponents which states that when multiplying like bases, you add the exponents. So:
[tex]\[
a^6 \times a^2 = a^{6+2} = a^8
\][/tex]
4.3.3: [tex]\((x^2)^3\)[/tex]
The power of a power property for exponents tells us that you multiply the exponents. Therefore:
[tex]\[
(x^2)^3 = x^{2 \times 3} = x^6
\][/tex]
4.3.5: [tex]\(\sqrt[3]{127 - 2} \times (-5)^2\)[/tex]
First, simplify inside the cube root:
[tex]\[
127 - 2 = 125
\][/tex]
Now, find the cube root of 125:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
Next, calculate [tex]\((-5)^2\)[/tex]:
[tex]\[
(-5)^2 = 25
\][/tex]
Finally, multiply the results:
[tex]\[
5 \times 25 = 125
\][/tex]
4.3.6: [tex]\(\frac{\sqrt{81}}{3} + \frac{\sqrt[3]{216}}{3}\)[/tex]
First, simplify [tex]\(\sqrt{81}\)[/tex]:
[tex]\[
\sqrt{81} = 9
\][/tex]
Divide by 3:
[tex]\[
\frac{9}{3} = 3
\][/tex]
Next, simplify [tex]\(\sqrt[3]{216}\)[/tex]:
[tex]\[
\sqrt[3]{216} = 6
\][/tex]
Divide by 3:
[tex]\[
\frac{6}{3} = 2
\][/tex]
Add the results:
[tex]\[
3 + 2 = 5
\][/tex]
The simplified results for each part are:
- 4.3.1: [tex]\(a^8\)[/tex]
- 4.3.3: [tex]\(x^6\)[/tex]
- 4.3.5: 125
- 4.3.6: 5
4.3.1: [tex]\(a^6 \times a^2\)[/tex]
To simplify this, we use the property of exponents which states that when multiplying like bases, you add the exponents. So:
[tex]\[
a^6 \times a^2 = a^{6+2} = a^8
\][/tex]
4.3.3: [tex]\((x^2)^3\)[/tex]
The power of a power property for exponents tells us that you multiply the exponents. Therefore:
[tex]\[
(x^2)^3 = x^{2 \times 3} = x^6
\][/tex]
4.3.5: [tex]\(\sqrt[3]{127 - 2} \times (-5)^2\)[/tex]
First, simplify inside the cube root:
[tex]\[
127 - 2 = 125
\][/tex]
Now, find the cube root of 125:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
Next, calculate [tex]\((-5)^2\)[/tex]:
[tex]\[
(-5)^2 = 25
\][/tex]
Finally, multiply the results:
[tex]\[
5 \times 25 = 125
\][/tex]
4.3.6: [tex]\(\frac{\sqrt{81}}{3} + \frac{\sqrt[3]{216}}{3}\)[/tex]
First, simplify [tex]\(\sqrt{81}\)[/tex]:
[tex]\[
\sqrt{81} = 9
\][/tex]
Divide by 3:
[tex]\[
\frac{9}{3} = 3
\][/tex]
Next, simplify [tex]\(\sqrt[3]{216}\)[/tex]:
[tex]\[
\sqrt[3]{216} = 6
\][/tex]
Divide by 3:
[tex]\[
\frac{6}{3} = 2
\][/tex]
Add the results:
[tex]\[
3 + 2 = 5
\][/tex]
The simplified results for each part are:
- 4.3.1: [tex]\(a^8\)[/tex]
- 4.3.3: [tex]\(x^6\)[/tex]
- 4.3.5: 125
- 4.3.6: 5
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