Problem 33 - College Algebra

Question

Simplify each exponential expression $$ \left(x^{-5}\right)^{3} $$

Step-by-Step Solution

Answer

\(x^{-15}\)

Step 1: Recognizing Power Of A Power Property

The problem \((x^{-5})^{3}\) can be seen as a base of \(x^{-5}\) raised to the power of \(3\). This forms the scenario of a power of a power, which states that \((a^{m})^{n}\) is equal to \(a^{m \times n}\).

Step 2: Apply The Power Of A Power Property

Applying the rule of a power of a power, we can simplify \((x^{-5})^{3}\) to \(x^{-5 \times 3}\).

Step 3: Multiply The Exponents

Multiplying the exponents \(-5 * 3\) results in -15.

Step 4: Final Answer

Putting all steps together, \((x^{-5})^{3}\) simplifies to \(x^{-15}\).

Key Concepts

Exponentiation Rules

Understanding exponentiation rules is essential for simplifying complex expressions. Exponentiation involves a base and an exponent. The base is the number being multiplied, while the exponent indicates how many times the base is used as a factor. For example, in the expression \(x^5\), \(x\) is the base and 5 is the exponent.
When working with exponential expressions, there are several key rules:

• Product of Powers rule: \(a^m \times a^n = a^{m+n}\)
• Power of a Power rule: \((a^m)^n = a^{m \times n}\)
• Power of a Product rule: \((ab)^n = a^n \times b^n\)
• Zero Exponent rule: \(a^0 = 1\) if \(a eq 0\)
• Negative Exponent rule: \(a^{-n} = \frac{1}{a^n}\)

Knowing these rules helps in rewriting and simplifying expressions, making complex calculations manageable. With these, you can confidently solve exercises like converting \((x^{-5})^3\).

Power of a Power Property

The Power of a Power property is a powerful tool in algebraic simplification. It comes into play when an exponential expression is raised to another power. This occurs in the form \((a^m)^n\). The rule states that you should multiply the exponents: \((a^m)^n = a^{m \times n}\).
This property simplifies calculations by reducing the number of operations required. Instead of repeatedly multiplying an expression, you calculate the product of the exponent values and rewrite the base raised to this new exponent.
In the exercise \((x^{-5})^{3}\), we apply this property by multiplying the inner exponent \(-5\) by the outer exponent \(3\), resulting in \(-15\). Thus, \((x^{-5})^{3}\) simplifies to \(x^{-15}\). Understanding this property is essential to handle complex exponentials efficiently.

Algebraic Simplification

Algebraic simplification involves writing an expression in its simplest form. It's crucial in math to make expressions easier to understand and work with. By simplifying, you reduce potential errors and can more easily compute values.
To simplify an expression with exponents, use the rules of exponentiation to eliminate extra terms and to simplify powers. This often involves:

• Applying exponent rules like the power of a power property
• Combining like terms
• Converting negative exponents to positive by using reciprocal values

Take the problem \((x^{-5})^{3}\). Through the power of a power rule, it simplifies to \(x^{-15}\). This is a cleaner expression and thus easier to interpret or compute further. Simplification makes problem-solving more efficient and clear, paving the way to analyze more complex algebraic operations.