Problem 80 - Elementary Algebra

Question

Find the value of \(\frac{2^{5}-4^{2}}{3^{-2}}\).

Step-by-Step Solution

Answer

Answer: The value of the expression is 144.

Step 1: Simplify the Terms

In this step, we will simplify the given expression. Calculate the values of each term in the expression: \(2^5\), \(4^2\), and \(3^{-2}\). \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\) \(4^2 = 4 \times 4 = 16\) \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\) Now, substitute these values back into the expression: \(\frac{2^{5}-4^{2}}{3^{-2}} = \frac{32-16}{\frac{1}{9}}\)

Step 2: Perform Subtraction

In this step, we will perform the subtraction in the numerator: \(\frac{32-16}{\frac{1}{9}} = \frac{16}{\frac{1}{9}}\)

Step 3: Divide by the Fraction

In this step, we will divide by the fraction in the denominator. Remember that dividing by a fraction is the same as multiplying by its reciprocal. \(\frac{16}{\frac{1}{9}}= 16 * \frac{9}{1} = 16 * 9\)

Step 4: Perform Multiplication

In this step, we will perform the multiplication: \(16 * 9= 144\) Thus, the value of the given expression \(\frac{2^{5}-4^{2}}{3^{-2}}\) is 144.

Key Concepts

Order of operations

In mathematics, the order of operations is crucial to solving expressions correctly. It is often remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. This sequence dictates the order in which operations should be performed to ensure accurate results.

When addressing the original exercise, one must first simplify terms involving exponents, as they take precedence over other operations like subtraction and division. Multiplication and division are performed from left to right, followed by addition and subtraction, also from left to right. In the given problem, \[ \frac{2^{5}-4^{2}}{3^{-2}} \], you'll simplify each component involving exponents before performing any subtraction or dealing with division by a fraction.

• Step 1: Solve the exponents separately, i.e., calculate \(2^5\), \(4^2\), and \(3^{-2}\).
• Step 2: Perform the subtraction in the numerator \(32-16\).
• Step 3: Manage the division by \(3^{-2}\) by converting it to multiplication by \(9\).

Following the order of operations ensures you systematically address each part, avoiding mistakes.

Simplifying expressions

Simplifying mathematical expressions is the process of transforming them into a form that's easier to interpret or calculate. This step is vital to making complex expressions more comprehensible, and it often involves performing operations or substitutions to reduce the complexity.

The given expression \(\frac{2^{5}-4^{2}}{3^{-2}}\) is simplified by evaluating the powers first. Each term with an exponent becomes a simple number, making the expression easier to work with. After resolving the exponents:

• \(2^5\) simplifies to 32.
• \(4^2\) simplifies to 16.
• \(3^{-2}\), which is a negative power, transforms into a fraction, simplifying to \(\frac{1}{9}\).

Once simplified, the expression \(\frac{32-16}{\frac{1}{9}}\) becomes easier to solve by following arithmetic operations. The end goal is to reach the simplest and most concise form of the expression possible.

Exponent rules

Understanding exponent rules is crucial for solving expressions involving powers. Exponents are shorthand notation for repeated multiplication of a number by itself. The rules governing exponents make calculations more accessible and allow for precise expression simplification.

Several key exponent rules are applied in the exercise:

• Power of a power: When a number is raised to an exponent, and then that result is raised to another exponent, you multiply the exponents. However, this is not directly used in this problem but is fundamental for understanding exponents.
• Negative exponents: These indicate reciprocals. For instance, \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).
• Zero exponents: Any non-zero number raised to the power of zero is equal to 1. This rule isn't directly applied here but is crucial for complex expressions.

Recognizing these exponent rules helps simplify and compute expressions efficiently. In our problem, applying these rules to \(3^{-2}\) was vital in transforming it into a reciprocal, facilitating straightforward multiplication in the final step.