Problem 54 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left[(-1.7)^{9}\right]^{8} $$

Step-by-Step Solution

Answer

The simplified expression is \\( (-1.7)^{72} \\\).

Step 1: Identify the Power Rule

The power rule for exponents states that when you raise a power to another power, you multiply the exponents. In general, \( (a^m)^n = a^{m \times n} \).

Step 2: Apply the Power Rule

Apply the power rule to the expression \( \left[(-1.7)^9\right]^8 \) by multiplying the exponents: 9 and 8.

Step 3: Multiply the Exponents

Multiply the exponents: \( 9 \times 8 = 72 \).

Step 4: Simplify the Expression

Using the result from the multiplication, simplify the expression to \( (-1.7)^{72} \). No need to calculate further as the question asks for the expression using exponents.

Key Concepts

Simplifying Expressions

When simplifying expressions, your goal is to make them easier to work with while retaining their original value. Think of it like organizing a messy drawer; you want to keep everything useful but make it tidy. In math, simplifying means reducing an expression to its most basic form without changing its value.

For expressions involving exponents, simplification often involves using rules like the power rule for exponents or other algebraic rules. Making expressions simpler helps you solve algebra problems more efficiently and avoid mistakes. Simplified expressions are more elegant and easier to understand.

Multiplying Exponents

Multiplying exponents can seem a bit tricky at first, but it actually follows a straightforward rule. This rule comes into play when you have an exponent raised to another exponent, like \[ (a^m)^n \] In such cases, you multiply the exponents together rather than the numbers themselves. This is known as the power rule for exponents.

• Identify the base and exponents in the expression.
• Multiply the exponents while keeping the base the same.

In our example, \[ \left[ (-1.7)^9 \right]^8 \]we recognize that -1.7 is the base, and we have 9 and 8 as the exponents. Multiplying them gives us \[ 9 \times 8 = 72 \] Thus, the simplified expression is \[ (-1.7)^{72} \] This simplification makes complex expressions much easier to comprehend and use in further calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators like addition or multiplication. They represent real-world situations in mathematical form, allowing us to solve problems systematically.

When working with exponents in algebraic expressions, understanding how to manipulate these components is crucial. The power rule for exponents is a valuable tool in this process, helping to simplify expressions quickly. Simplification can make equations easier to solve or interpret by reducing the complexity of the expression.

In broader terms, mastering how to manage algebraic expressions with exponents allows for deeper exploration in fields like physics and engineering. This foundational skill is essential for any math student and helps build confidence in handling more sophisticated problems involving polynomials or exponential functions.