Write the following without negative exponents: (a^-1 + b^-1)^-1
(a^-1 + b^-1)^-1
= (1/a + 1/b)^-1
=((b+a)/(ab))^-1
= (ab)/(a+b)
(1/a + 1/b)^-1 = 1 / (1/a + 1/b).
To write the expression (a^-1 + b^-1)^-1 without negative exponents, we can make use of the exponent properties.
First, let's rewrite the expression with positive exponents:
(a^-1 + b^-1)^-1
Using the property that a^-m = 1/a^m, we can rewrite the exponents so that they become positive:
(1/a + 1/b)^-1
Next, we need to evaluate the expression inside the parentheses by finding a common denominator for a and b. The common denominator is ab, so we can rewrite 1/a and 1/b with a common denominator:
(1/a + 1/b)^-1 = (b/ab + a/ab)^-1
Notice that we have a common denominator of ab now. We can combine the terms inside the parentheses:
= (b + a)/(ab)^-1
= (b + a)/(1/ab)
Finally, to simplify further, we can multiply the numerator and denominator by ab:
= (b + a)(ab)/(1)
= ab(b + a)
So, without negative exponents, the expression (a^-1 + b^-1)^-1 simplifies to ab(b + a).