^ = to the power
1. 8^1/2*8^-5/2
2.3^5/3(divided by) 3^2/3
**please show work
to multiply add exponents if the base is the same
8^(1/2 -5/2)
= 8^(-2)
= 1/8^2
= 1/64
1/3^(2/3) = 3^(-2/3)
so
=3^(5/3) * 3^(-2/3)
=3^(3/3)
=3
i do not understand the second problem.
negative exponent definition:
a^-b = 1/(a^b)
derivation:
when you multiply you add exponents
when you divide you subtract exponents (remember a^0 = 1)
1/a^b = a^0/a^b = a^(0-b) = a^-b
Certainly! I'll show you how to solve each problem step by step.
1. 8^(1/2) * 8^(-5/2)
To simplify this expression, you can use the rule of exponents that states a^m * a^n = a^(m+n). Applying this rule,
8^(1/2) * 8^(-5/2) = 8^[(1/2) + (-5/2)]
Next, you can combine the exponents:
[(1/2) + (-5/2)] = -4/2 = -2
Therefore, the expression simplifies to:
8^(-2)
Now, using the rule that a^(-n) = 1 / (a^n), we can rewrite the expression as:
1 / 8^2
Simplifying further:
1 / 64
So, the answer is 1/64.
2. (3^(5/3)) / (3^(2/3))
To divide two exponential expressions with the same base, you can subtract the exponents. Using this rule:
(3^(5/3)) / (3^(2/3)) = 3^[(5/3) - (2/3)]
Next, subtracting the exponents:
[(5/3) - (2/3)] = 3/3 = 1
Therefore, the expression simplifies to:
3^1
Since any number raised to the power of 1 is itself, the answer is:
3
So, the answer is 3.
I hope that helps! Let me know if you have any further questions.