Problem 53 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left[(-4.3)^{3}\right]^{8} $$

Step-by-Step Solution

Answer

The simplified expression is \((-4.3)^{24}\).

Step 1: Understanding the Power Rule for Exponents

The power rule for exponents states that when you raise a power to another power, you multiply the exponents. In mathematical terms: \[(a^m)^n = a^{m imes n}\]Here, we apply this rule to simplify the given expression.

Step 2: Identify the Base and Exponents

In the expression \( \left[(-4.3)^{3}\right]^{8} \), the base is \(-4.3\), the first exponent \( m \) is 3, and the second exponent \( n \) is 8.

Step 3: Apply the Power Rule

Using the power rule \((a^m)^n = a^{m \times n}\), apply it to the expression. Multiply the exponents:\[ (-4.3)^{3 \times 8} = (-4.3)^{24} \]

Step 4: Write the Simplified Expression

The expression \( \left[(-4.3)^{3}\right]^{8} \) simplifies to \((-4.3)^{24}\) when you apply the power rule for exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex expressions easier to work with. By reducing them to their simplest form, math problems become less daunting.
This often involves performing operations like combining like terms or using mathematical rules. In this exercise, the power rule for exponents is crucial. Simplification can make it easier to recognize the basic elements in a problem and see how they relate.

• Break down the problem.
• Use known rules to simplify.
• Recheck your steps to confirm accuracy.

For instance, simplifying \( \left[(-4.3)^{3}\right]^{8} \) makes it less complicated and leads us to easier calculations in future steps.

Exponents

Exponents are a mathematical notation indicating the number of times a number is multiplied by itself. When you see an expression like \(a^n\), it means that the number \a\, known as the base, is multiplied by itself \ times.
Understanding exponents is important because they're present in many mathematical contexts, including growth calculations in finance and science.

• The base is the number being multiplied.
• The exponent shows how many times to multiply the base by itself.

In this exercise, \(-4.3\)\ is the base and \ 3\ and \ 8\ are exponents. Using the power rule helps manage exponents efficiently by allowing multiplication when raising one exponent to another.

Mathematical Properties

Mathematical properties are rules and laws that allow us to manipulate and simplify expressions. They include properties like commutative, associative, and distributive laws. For exponents especially, properties like the power rule are essential.

• The Power Rule: \((a^m)^n = a^{m \times n}\)
• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

These properties allow us to tackle and simplify mathematical expressions efficiently. With the power rule, we multiply the exponents, in this case resulting in \(-4.3 \) raised to the 24th power. Understanding these rules makes math more manageable and helps solve problems systematically.