Question

Simplify the expression. Write your answer using only positive exponents. \(\left(z^2\right)^{-3}\)

Short answer

The simplified form of the given expression using only positive exponents is \( 1/z^6 \).

Step-by-step solution

Apply the Rule

Apply the rule of exponents \( (a^m)^n = a^{m*n} \). Here, \( a=z \), \( m=2 \), and \( n=-3 \). As a result, we get: \( (z^2)^{-3} = z^{2*-3} \).

Simplify the Expression

Now we simply have to multiply the exponents to simplify the expression. Hence, \( z^{2*-3} = z^{-6} \).

Write with positive exponent

The exercise specifies to write the answer with positive exponent. We can convert the negative exponent to a positive one by taking the reciprocal. Thus, the expression now becomes \( z^{-6} = 1/z^6 \).

Key concepts

Understanding Negative Exponents

When working with negative exponents, a common source of confusion stems from understanding what they represent. A negative exponent indicates that the base number is on the wrong side of a fraction and needs to be flipped to the other side. For instance, in the expression\( a^{-n} \), the negative exponent \( -n \) tells us to take the reciprocal of the base \( a \) and then apply the positive exponent. In other words, \( a^{-n} = \frac{1}{a^n} \).

This concept is crucial because it simplifies expressions that might otherwise look complex. For students to fully grasp negative exponents, it's essential to practice converting them into their equivalent positive forms. By repeatedly using this rule, the process becomes intuitive, allowing for a quick and accurate simplification of expressions.

Applying Exponent Rules

To simplify exponential expressions effectively, mastering the exponent rules is imperative. These rules are the shortcuts that allow us to manipulate exponents quickly. For example, the rule \( (a^m)^n = a^{m*n} \) used in the original exercise is known as the 'power of a power' rule. It states that when raising a power to another power, you multiply the exponents.

Other important rules include the 'product of powers' rule, which states that when multiplying like bases, you add the exponents \( (a^m * a^n = a^{m+n}) \), and the 'quotient of powers' rule, which means when dividing like bases, you subtract the exponents \( (\frac{a^m}{a^n} = a^{m-n}) \). Understanding and applying these rules is fundamental to working efficiently with exponents in algebra.

Writing Exponents Positively

In mathematics, it's often required to write expressions with only positive exponents. To do this with an expression that has a negative exponent, we use the rule for negative exponents which states that any base with a negative exponent can be rewritten as the reciprocal with a positive exponent, as seen in the solution step \( z^{-6} = \frac{1}{z^6} \).

This process of writing exponents positively does not change the value of the expression but rather presents it in a form that is often more useful, especially in further calculations or when applying additional algebraic operations. It's a simple yet powerful technique that enhances the readability and simplification of complex expressions.