Answer :
Alright, let's analyze each given expression and match it with the correct method needed to evaluate it.
1. \((4x^3)^5\)
- The expression is of the form \((a \cdot x^n)^m\). To simplify this, you need to multiply the exponents inside the parentheses by the outer exponent.
[tex]\[ (4x^3)^5 = 4^5 \cdot (x^3)^5 = 4^5 \cdot x^{3 \cdot 5} = 4^5 \cdot x^{15} \][/tex]
- Method: Multiply the exponents.
2. \(5^3 \cdot 5^3\)
- The expression is a product of powers with the same base. When multiplying exponents with the same base, you add the exponents.
[tex]\[ 5^3 \cdot 5^3 = 5^{3+3} = 5^6 \][/tex]
- Method: Add the exponents.
3. \((7^2)^3\)
- The expression is a power of a power. To simplify this, you multiply the exponents.
[tex]\[ (7^2)^3 = 7^{2 \cdot 3} = 7^6 \][/tex]
- Method: Multiply the exponents.
4. \(6^9 + 6^5\)
- This expression involves adding terms with the same base but different exponents. Exponent rules for multiplication or division do not apply here, and the expression cannot be simplified using exponent rules.
[tex]\[ 6^9 + 6^5 \][/tex]
- Method: Cannot be simplified using exponent rules.
By evaluating each expression and matching it with the correct method, we have:
- \((4x^3)^5 \rightarrow\) Multiply the exponents.
- \(5^3 \cdot 5^3 \rightarrow\) Add the exponents.
- \((7^2)^3 \rightarrow\) Multiply the exponents.
- [tex]\(6^9 + 6^5 \rightarrow\)[/tex] Cannot be simplified using exponent rules.
1. \((4x^3)^5\)
- The expression is of the form \((a \cdot x^n)^m\). To simplify this, you need to multiply the exponents inside the parentheses by the outer exponent.
[tex]\[ (4x^3)^5 = 4^5 \cdot (x^3)^5 = 4^5 \cdot x^{3 \cdot 5} = 4^5 \cdot x^{15} \][/tex]
- Method: Multiply the exponents.
2. \(5^3 \cdot 5^3\)
- The expression is a product of powers with the same base. When multiplying exponents with the same base, you add the exponents.
[tex]\[ 5^3 \cdot 5^3 = 5^{3+3} = 5^6 \][/tex]
- Method: Add the exponents.
3. \((7^2)^3\)
- The expression is a power of a power. To simplify this, you multiply the exponents.
[tex]\[ (7^2)^3 = 7^{2 \cdot 3} = 7^6 \][/tex]
- Method: Multiply the exponents.
4. \(6^9 + 6^5\)
- This expression involves adding terms with the same base but different exponents. Exponent rules for multiplication or division do not apply here, and the expression cannot be simplified using exponent rules.
[tex]\[ 6^9 + 6^5 \][/tex]
- Method: Cannot be simplified using exponent rules.
By evaluating each expression and matching it with the correct method, we have:
- \((4x^3)^5 \rightarrow\) Multiply the exponents.
- \(5^3 \cdot 5^3 \rightarrow\) Add the exponents.
- \((7^2)^3 \rightarrow\) Multiply the exponents.
- [tex]\(6^9 + 6^5 \rightarrow\)[/tex] Cannot be simplified using exponent rules.
