Fractional Indices

Question

Question 1:

8^1/6×2^x=32^1/2

Find x

Question 2:

5^-2×3 Square root 8

As a decimal

Answer 1

8^ (1/6) * 2^x = 32^(1/2)

[2^(3)]^(1/6) * 2^x = (2^5)^(1/2) <---- 8 = 2^3 and 32 = 2^5

2^(3/6) * 2^x = 2^(5/2) <---- properties of exponents: monomial to power, so multiplies exponents

2^(1/2)*2^x = 2^(5.2) <--- 3/6 reduces to 1/2

2^(x+1/2) = 2^(5/2) <--- properties of exponents: multiplies monomials with same base, adds exponents

x+1/2 = 5/2

x = 5/2-1/2 = 4/2 = 2

======================================

5^-2×3 Square root 8 =

1/(5^2) * 3 sqrt(8) <---- rewrite with positive exponent

=1/25 * 3 * sqrt(8)

= 3/25 * sqrt(8)

= 3/25 * sqrt(2*2*2)

= 3/25 * 2 * sqrt(2)

= 6/25 * sqrt(2)

which is approximately 0.33941125496954281171240529381033......