Problem 48 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \left(x^{3}\right)^{-2} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression, \(\left(x^{3}\right)^{-2}\), using only positive exponents. Answer: \(\frac{1}{x^6}\)

Step 1: Understand the exponent rule for powers of powers

When a power is raised to another power, we multiply the exponents. For example, \((a^m)^n = a^{mn}\). In this case, we want to simplify \(\left(x^{3}\right)^{-2}\).

Step 2: Apply the exponent rule to simplify the expression

To remove the negative exponent, we have to apply the exponent rule for powers of powers. So, we'll multiply the exponents: \(\left(x^{3}\right)^{-2} = x^{3 \cdot (-2)}\)

Step 3: Calculate the new exponent

Now, we need to calculate the new exponent by multiplying 3 and (-2): \(3 \cdot (-2) = -6\) This gives us: \(x^{-6}\)

Step 4: Remove the negative exponent

To rewrite the expression using only positive exponents, we need to remove the negative exponent. To do that, we'll use the following rule: \(x^{-n} = \frac{1}{x^n}\) So, we'll rewrite \(x^{-6}\) as: \(x^{-6} = \frac{1}{x^6}\)

Step 5: Write the final result

Now that we've rewritten the expression with positive exponents, we have our final result: \(\left(x^{3}\right)^{-2} = \frac{1}{x^6}\)

Key Concepts

Positive Exponents

When we work with exponents, there is a simple rule to remember for the sake of clarity and positivity: any exponent can be positive. Positive exponents are straightforward because they represent how many times a base number is multiplied by itself. For example, if you have \(x^3\), it means you multiply \(x\) by itself three times: \(x \times x \times x\). Positive exponents help in representing large numbers in a concise way.

When writing expressions with exponents, expressing them positively is often more practical and universally understandable in mathematical problems. This is because it avoids complications that arise with interpreting negative exponents, which we will address shortly. Therefore, converting expressions to positive exponents, as practiced in problems like the one with \(\left(x^{3}\right)^{-2}\), is crucial in mathematical simplification tasks.

Negative Exponents

Negative exponents might seem intimidating at first, but they are quite handy once you understand them. Essentially, they represent the reciprocal of the base raised to the corresponding positive exponent. This is the primary rule for negative exponents: \(x^{-n} = \frac{1}{x^n}\). This means that instead of decreasing the power by one, a negative exponent inverts the base variable.

For instance, in the exercise \(x^{-6}\), the negative exponent of \(-6\) indicates that you take the reciprocal of \(x^6\), rewriting it as \(\frac{1}{x^6}\). This flips the position of \(x^6\) to the denominator of a fraction.

• Negative exponents transform the expression to its reciprocal.
• They make division simpler because they turn division operations into multiplications.

Practicing this method helps in efficiently converting negative exponent expressions into an easier-to-understand format.

Powers of Powers Rule

In mathematics, handling powers of powers is a must-know skill. This rule involves raising a power to another power, and it has a straightforward method: multiply the exponents. When you come across an expression like \((x^3)^{-2}\), the power of \(x\) is initially set to 3, but it is soon to be influenced by the outer exponent \(-2\).

Using the rule that \((a^m)^n = a^{mn}\), you multiply the inner exponent \(3\) by the outer exponent \(-2\):

• Expression: \((x^3)^{-2}\)
• Multiply exponents: \(3 \cdot (-2) = -6\)

After following through with the multiplication, you find the expression reduced to \(x^{-6}\).

This powerful technique provides an efficient way to handle exponents that are layered or nested, simplifying them through multiplication and then following through with the application of rules like those for negative exponents. Practice and attention to these simple steps can help readily prepare you for handling any exponent situation with ease!