91Ó°ÊÓ

When we have an expression with a power raised to another power, we use the power of a power rule to simplify it. This rule states that when you raise a power to another power, you multiply the exponents. So, for an expression like \( (a^m)^n \), you multiply the exponent \(m\) by the exponent \(n\) to get \( a^{m*n} \).

Let's apply this rule to an exercise to see it in action. If we have \( (2^{-2})^x \) and we need to simplify it, we multiply the exponents -2 (from the inner power) and \(x\) (from the outer power), getting \( 2^{-2x} \). This process simplifies complex expressions and makes them easier to work with when solving equations.

Exponents are a shorthand way to express repeated multiplication. When we write \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times. For instance, \(2^3 = 2 \times 2 \times 2\). An important aspect of exponents to remember is their behavior in different scenarios—when negative or as fractions. For example, a negative exponent like \(2^{-2}\) implies \(\frac{1}{2^2}\), or \(\frac{1}{4}\). Understanding the behavior of exponents is vital when we encounter them in equations we need to solve.

To solve exponent equations efficiently, it's often helpful to have the same base on both sides of the equation. This is because we can then set the exponents equal to each other, which simplifies the process considerably. For example, if we have an equation \(2^{a} = 2^{b}\), we can assume that \(a = b\) based on the property that if the bases are equal and the quantities are equal, then the exponents must also be equal.

Applying this to our original exercise \(\left(\frac{1}{4}\right)^x = 64\), we express \(\frac{1}{4}\) and 64 as \(2^{-2}\) and \(2^6\), respectively, setting up the equation with the same base. When we have \(2^{-2x} = 2^6\), we can directly set \( -2x = 6\) because the bases (2) are identical, leading us to the answer \(x = -3\).