Question

Rewrite the expression with positive exponents and simplify.\(\left(\frac{2 z^{2}}{y}\right)^{-2}\)

Short answer

The expression \(\left(\frac{2z^{2}}{y}\right)^{-2}\) rewritten with positive exponents and simplified is \(\frac{y^{2}}{4z^{4}}\).

Step-by-step solution

Apply Power of a Quotient Rule

The power of a quotient rule states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Apply this rule to \(\left(\frac{2z^{2}}{y}\right)^{-2}\) to separate the numerator from the denominator. This gives \(\frac{(2z^{2})^{-2}}{y^{-2}}\).

Apply Power of a Power Rule

Apply the power of a power rule, which states that \((a^n)^m = a^{nm}\) to rewrite the numerator. This gives \(\frac{2^{-2}z^{-4}}{y^{-2}}\).

Simplifying the Expression

Remember, the negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Apply this rule to invert all terms with negative exponents. This gives \(\frac{1}{2^{2}z^{4}} * y^{2}\). Further simplify it to \(\frac{y^{2}}{4z^{4}}\).

Result

The expression \(\left(\frac{2z^{2}}{y}\right)^{-2}\) rewritten with positive exponents and simplified is \(\frac{y^{2}}{4z^{4}}\).

Key concepts

Power of a Quotient Rule

Understanding the power of a quotient rule is essential in algebra, especially when dealing with exponents. This rule comes into play when you have a fraction raised to a power. In simple terms, if you have \(\left(\frac{a}{b}\right)^n\), you can apply the exponent to both the numerator and the denominator separately. It transforms the expression into \(\frac{a^n}{b^n}\).

Let's see how this applies to our exercise. When we have \(\left(\frac{2z^{2}}{y}\right)^{-2}\), using the power of a quotient rule allows you to separate the power to both the numerator - \(2z^{2}\) - and the denominator - \(y\), resulting in \(\frac{(2z^{2})^{-2}}{y^{-2}}\). This step is crucial because it sets the stage for further simplification under the standard exponent rules. It's like unpacking a complex expression into smaller, more manageable pieces that follow the same mathematical logic. This rule helps maintain order and clarity in the simplification process.

Power of a Power Rule

The power of a power rule takes the concept of exponents one step further. When an expression already contains an exponent and is then raised to another power, this rule allows simplification. Mathematically, it's shown as \( (a^n)^m = a^{nm} \). This compact notation is all about multiplication - you multiply the exponents.

Applying this to the numerator in our example \(2^{-2}z^{-4}\), we multiplied the exponent of the number 2 and the variable \(z\) by the outside exponent \( -2 \). This step is important to understand because it prevents errors when dealing with layers of exponents. Always remember to multiply the exponents, it's a common pitfall to mistakenly add them instead. Multiply the exponents across the entire expression to proceed to the next step in simplification correctly.

Simplifying Algebraic Expressions

Simplifying algebraic expressions can sometimes feel like a puzzle. The goal is to rewrite the expression in the simplest form possible. Negative exponents often complicate things but worry not, they follow a simple rule: \(a^{-n} = \frac{1}{a^n}\). To simplify, you turn a negative exponent into a positive one by flipping the base to the other side of the fraction.

In the final steps of our problem, applying the negative exponent rule to \(2^{-2}\) and \(z^{-4}\) in the numerator allows us to rewrite them as \(\frac{1}{2^2}\) and \(\frac{1}{z^4}\), respectively. The term \(y^{-2}\) in the denominator becomes \(y^{2}\) in the numerator upon inversion. Simplifying further, we combine these to get the final expression \(\frac{y^{2}}{4z^{4}}\), which much more neatly represents the original complex expression. This step-by-step method ensures that every part of the expression is addressed and properly transformed, enhancing a student's grasp of the process involved in simplifying algebraic expressions with exponents.