Problem 51 - Elementary Algebra

Question

\(\left(y^{-3}\right)^{2}\)

Step-by-Step Solution

Answer

y^{-6}

Step 1: Understand the given expression

The given expression is \(\backslash left(y^{-3}\backslash right)^{2}\). This means raising \(y^{-3}\) to the power of 2.

Step 2: Use the power of a power rule

According to the rule \[ (a^m)^n = a^{m \times n} \], we can simplify \( \backslash left( y^{-3} \backslash right)^{2} \) as \( y^{(-3) \times 2} \).

Step 3: Perform the multiplication

Multiply the exponents: \[ (-3) \times 2 = -6 \]. Thus, we get \( y^{-6} \).

Key Concepts

Power of a Power Rule

To understand the concept of the 'power of a power rule', let's break it down. This rule applies when you have an exponent raised to another exponent. The rule is written as \( (a^m)^n = a^{m \times n} \). In simpler terms, you multiply the exponents.
For example:

• If you have \( (x^2)^3 \), you multiply 2 and 3 to get 6. So, \( (x^2)^3 = x^6 \).

In our exercise, we have \( (y^{-3})^2 \). Using the rule, we multiply the exponents \( -3 \) and \( 2 \). This becomes \( y^{(-3) \times 2} = y^{-6} \).

Negative Exponents

Negative exponents can seem tricky, but they are straightforward when you know the rule. A negative exponent means that the base should be taken into the denominator and the exponent becomes positive.
For example:

• \( x^{-2} \) means \( \frac{1}{x^2} \).

In our exercise, after applying the power of a power rule, we got \ y^{-6} \. This can be rewritten as \ \frac{1}{y^6} \. Remember, moving the exponent from the numerator to the denominator changes its sign.

Simplifying Exponents

Simplifying exponents is all about making expressions easier to work with.
Here are some quick tips for simplifying:

• Use the 'power of a power' rule: Multiply the exponents.
• Observe the sign of the exponents. If they are negative, consider using the rule for negative exponents.
• Transform negative exponents into positive ones, by flipping the base into a denominator.

In our problem, after simplification, we've taken \( (y^{-3})^2 \) and transformed it into \ y^{-6} \ which can then be rewritten as \ \frac{1}{y^6} \. This makes the expression much easier to understand and work with.