Question
(29)½*(29⁵)³/² as indices
Answers
Answered by
GPT-4o mini
To simplify the expression \((29)^{\frac{1}{2}} \cdot (29^5)^{\frac{3}{2}}\) using the laws of indices, we can apply the following steps:
1. **Apply the power of a power rule:** When raising a power to a power, you multiply the exponents. Thus, for the second term \((29^5)^{\frac{3}{2}}\), we have:
\[
(29^5)^{\frac{3}{2}} = 29^{5 \cdot \frac{3}{2}} = 29^{\frac{15}{2}}.
\]
2. **Combine the powers:** Now we have:
\[
(29)^{\frac{1}{2}} \cdot (29)^{\frac{15}{2}}.
\]
Using the rule for multiplying powers with the same base (add the exponents), we have:
\[
29^{\frac{1}{2} + \frac{15}{2}} = 29^{\frac{1 + 15}{2}} = 29^{\frac{16}{2}} = 29^8.
\]
Therefore, the simplified expression is:
\[
\boxed{29^8}.
\]
1. **Apply the power of a power rule:** When raising a power to a power, you multiply the exponents. Thus, for the second term \((29^5)^{\frac{3}{2}}\), we have:
\[
(29^5)^{\frac{3}{2}} = 29^{5 \cdot \frac{3}{2}} = 29^{\frac{15}{2}}.
\]
2. **Combine the powers:** Now we have:
\[
(29)^{\frac{1}{2}} \cdot (29)^{\frac{15}{2}}.
\]
Using the rule for multiplying powers with the same base (add the exponents), we have:
\[
29^{\frac{1}{2} + \frac{15}{2}} = 29^{\frac{1 + 15}{2}} = 29^{\frac{16}{2}} = 29^8.
\]
Therefore, the simplified expression is:
\[
\boxed{29^8}.
\]
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