Please Simplify:
(root5 a^7/2)^4 a^3/a^8
I am having a hard time figuring where to start on this problem. Can someone please help me?
(sqrt5)^4=25
(a^7/2)^4=(a^3.5)^4=a^14
a^14 *a^3*a^-8=a^9
check my thinking.
Not too sure if that is correct. This is what I got.
= (sqrt (5)4*(a (7/2)) 4) a3-8
=5(1/2)*4*a (7/2)*4*a3-8
=25 a14 a-5
=25 a9
Is this correct?
To simplify the given expression `(√5a^(7/2))^4 * a^3/a^8`, let's break it down step by step:
Step 1: Simplify the expression inside the parentheses `√5a^(7/2)`.
- The square root of a number is the same as raising it to the power of 1/2, so √5 = 5^(1/2).
- The exponent property states that when raising a number to a power, you need to multiply the exponents. Therefore, `a^(7/2) = a^7 * a^(1/2)`.
- Combining these rules, `√5a^(7/2) = 5^(1/2) * a^7 * a^(1/2) = 5^(1/2) * a^(7 + 1/2) = 5^(1/2) * a^(15/2)`.
Step 2: Simplify the expression `5^(1/2) * a^(15/2))^4 * a^3/a^8`.
- Applying the exponent rule, `(5^(1/2))^4 = 5^(1/2 * 4) = 5^2 = 25`.
- To divide two numbers with the same base, subtract the exponents, so `a^(15/2) / a^8 = a^(15/2 - 8) = a^(15/2 - 16/2) = a^(-1/2)`.
- Combining all the simplified terms, the expression becomes `25 * a^(-1/2) * a^3/a^8`.
Step 3: Simplify the expression further by combining the terms with the same base, 'a':
- When multiplying two terms with the same base, add the exponents. Thus, `25 * a^(-1/2) * a^3 = 25 * a^(3 - 1/2) = 25 * a^(5/2)`.
- Similarly, to divide two terms with the same base, subtract the exponents. Thus, `a^(5/2) / a^8 = a^(5/2 - 8) = a^(-7/2)`.
Step 4: Combine the remaining terms `25 * a^(-7/2)` with `a^(-7/2)`.
- When multiplying two terms with the same base, just multiply the coefficients. Therefore, `25 * a^(-7/2) * a^(-7/2) = 25 * a^(-7/2 + -7/2) = 25 * a^(-14/2) = 25 * a^(-7) = 25/a^7`.
Hence, the simplified expression is `25/a^7`.