Exponential Functions:
How do I work out something like (2u^3 v^-1)^-4 ? It’s really confusing for me, and I have a test tomorrow, but not even my teacher can help me understand. Please help!
(The ^ means exponent)
the exponent outside the parentheses gets "distributed" over every quantity inside
in this case ... 2^-4 * u^-12 * v^4 ... v^4 / (16 u^12)
To simplify the expression (2u^3 v^-1)^-4, you can follow these steps:
Step 1: Deal with the exponents inside the parentheses.
- For "u^3," keep the base "u" the same and multiply the exponents: u^(3 * -4) = u^(-12).
- For "v^-1," move the base "v" to the denominator and change the sign of the exponent: 1/v^1 = v^(-1).
Step 2: Rewrite the expression with the simplified exponents.
- (2u^3 v^-1)^-4 = (2u^-12 v^-1)^-4.
Step 3: Apply the negative exponent rule. In this case, it means moving the bases with negative exponents to the opposite side of the fraction line.
- (2u^-12 v^-1)^-4 = (2 / (u^12 * v))^-4 = 1 / (2u^12 v)^4.
Therefore, the expression (2u^3 v^-1)^-4 simplifies to 1 / (2u^12 v)^4.
To work out an expression like (2u^3v^-1)^-4, you need to apply the exponent to each part of the expression separately. Let's break down the steps:
Step 1: Apply the outermost exponent.
The outermost exponent is -4, so we need to apply it to the entire expression. This means we will raise the whole expression to the power of -4.
Step 2: Apply the exponent to each part separately.
The expression (2u^3v^-1) consists of two parts: 2u^3 and v^-1. We need to apply the exponent to each part individually.
For 2u^3, raise the base (2u^3) to the power of -4. This can be done by taking the reciprocal of the base and raising it to the power of 4.
(2u^3)^-4 = (1 / 2u^3)^4 = 1^4 / (2u^3)^4 = 1 / (2^4 * (u^3)^4) = 1 / (16u^12).
For v^-1, raise the base (v^-1) to the power of -4. This can be done by taking the reciprocal of the base and raising it to the power of 4.
(v^-1)^-4 = (1 / v^-1)^4 = 1^4 / (v^-1)^4 = 1 / (v^-4) = 1 / (1 / v^4) = v^4.
Step 3: Combine the results.
After applying the exponent to each part separately, we can combine the results. In this case, we obtained 1 / (16u^12) for the first part and v^4 for the second part.
Therefore, the final result is (1 / (16u^12)) * v^4, or v^4 / (16u^12).
Remember, when working with exponents, it's essential to follow the order of operations and apply the exponent to each part separately to arrive at the correct answer.