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Answer :
Sure! Let's simplify each expression step by step:
### 4.3.1 Simplify [tex]\( a^6 \times a^2 \)[/tex]
- When you multiply expressions with the same base, you add their exponents. So, [tex]\( a^m \times a^n = a^{m+n} \)[/tex].
- Here, we have [tex]\( a^6 \times a^2 \)[/tex].
- Add the exponents: [tex]\( 6 + 2 = 8 \)[/tex].
- Therefore, [tex]\( a^6 \times a^2 = a^8 \)[/tex].
### 4.3.2 Simplify [tex]\( z^4 \div z^6 \)[/tex]
- When you divide expressions with the same base, you subtract the exponents. So, [tex]\( a^m \div a^n = a^{m-n} \)[/tex].
- Here, we have [tex]\( z^4 \div z^6 \)[/tex].
- Subtract the exponents: [tex]\( 4 - 6 = -2 \)[/tex].
- Therefore, [tex]\( z^4 \div z^6 = z^{-2} \)[/tex].
### 4.3.3 Simplify [tex]\( \left(x^2\right)^3 \)[/tex]
- When raising a power to another power, you multiply the exponents. So, [tex]\( (a^m)^n = a^{m \times n} \)[/tex].
- Here, we have [tex]\( (x^2)^3 \)[/tex].
- Multiply the exponents: [tex]\( 2 \times 3 = 6 \)[/tex].
- Therefore, [tex]\( (x^2)^3 = x^6 \)[/tex].
In summary:
- [tex]\( a^6 \times a^2 = a^8 \)[/tex]
- [tex]\( z^4 \div z^6 = z^{-2} \)[/tex]
- [tex]\( (x^2)^3 = x^6 \)[/tex]
### 4.3.1 Simplify [tex]\( a^6 \times a^2 \)[/tex]
- When you multiply expressions with the same base, you add their exponents. So, [tex]\( a^m \times a^n = a^{m+n} \)[/tex].
- Here, we have [tex]\( a^6 \times a^2 \)[/tex].
- Add the exponents: [tex]\( 6 + 2 = 8 \)[/tex].
- Therefore, [tex]\( a^6 \times a^2 = a^8 \)[/tex].
### 4.3.2 Simplify [tex]\( z^4 \div z^6 \)[/tex]
- When you divide expressions with the same base, you subtract the exponents. So, [tex]\( a^m \div a^n = a^{m-n} \)[/tex].
- Here, we have [tex]\( z^4 \div z^6 \)[/tex].
- Subtract the exponents: [tex]\( 4 - 6 = -2 \)[/tex].
- Therefore, [tex]\( z^4 \div z^6 = z^{-2} \)[/tex].
### 4.3.3 Simplify [tex]\( \left(x^2\right)^3 \)[/tex]
- When raising a power to another power, you multiply the exponents. So, [tex]\( (a^m)^n = a^{m \times n} \)[/tex].
- Here, we have [tex]\( (x^2)^3 \)[/tex].
- Multiply the exponents: [tex]\( 2 \times 3 = 6 \)[/tex].
- Therefore, [tex]\( (x^2)^3 = x^6 \)[/tex].
In summary:
- [tex]\( a^6 \times a^2 = a^8 \)[/tex]
- [tex]\( z^4 \div z^6 = z^{-2} \)[/tex]
- [tex]\( (x^2)^3 = x^6 \)[/tex]
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