Problem 20 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(z^{2}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified form is \(z^{10}\).

Step 1: Identify the Power Rule for Exponents

When you have an expression of the form \(a^m\)^n, you multiply the exponents: this is the power rule for exponents. Using this rule, \(a^{m imes n}\) becomes the simplified form.

Step 2: Apply the Power Rule to Simplify the Expression

For the given expression \(\left(z^2\right)^5\), identify the base \(z\) with its exponent \(2\). Multiply this exponent by the outer exponent \(5\) to get the new exponent. This results in \(z^{2 \times 5} = z^{10}\).

Key Concepts

Exponents

Exponents, also known as powers, are a shorthand way of expressing repeated multiplication of the same number. If you see something like \(a^b\), it means that the base \(a\) is multiplied by itself \(b\) times. This concept is fundamental in mathematics as it helps in simplifying expressions and solving equations.
For example, \(2^3 = 2 \times 2 \times 2 = 8\). In this example:

• \(2\) is the base
• \(3\) is the exponent
• \(8\) is the result of the multiplication

Using exponents makes calculations quicker and more efficient. It allows us to express very large or very small numbers in a more concise form, which is especially useful in scientific notation.

Simplifying Expressions

Simplifying expressions is about making them as clean and efficient as possible, by reducing them to their simplest form. This often means combining like terms or applying mathematical rules like the power rule for exponents. Simplifying can make expressions easier to work with and solve.
Take the example from the original exercise: \(\left(z^2\right)^5\). By applying the power rule for exponents, you simplify it to \(z^{10}\). This makes the expression easier to handle in further mathematical operations.

• Combining like terms
• Using rules, such as the power rule, to reduce the complexity
• Makes calculations more straightforward and manageable

Remember, the goal of simplifying is to make expressions look neater without changing their value. This is achieved by following consistent mathematical procedures.

Multiplying Exponents

When you come across a situation where you need to multiply exponents, like \((a^m)^n\), it's important to apply the correct rules to simplify it correctly. This is where the power rule for exponents comes into play.
The power rule states that to simplify an expression of the form \((a^m)^n\), you multiply the exponents: \(a^{m \times n}\). Let's break it down using the exercise we have:

• The base is \(z\), which is being raised to the power \(2\) inside the parenthesis.
• This entire expression is then raised to the power of \(5\).
• According to the power rule, you multiply the exponents: \(z^{2 \times 5} = z^{10}\).

This powerful rule simplifies complex exponent expressions and helps perform calculations in a more streamlined and error-free manner.