Problem 93 - Algebra 2

Question

Which number is closest to \(\left(81 n^{2}\right)^{0.75}\) for \(n=2 ?\) \(\begin{array}{llll}{\text { E. } 45.4} & {\text { G. } 76.4} & {\text { H. } 243.0} & {\text { 1.2061.9 }}\end{array}\)

Step-by-Step Solution

Answer

The number closest to \((81 n^{2})^{0.75}\) for \(n=2\) is \(G: 76.4\).

Step 1: Substitute the value of n

First, substitute the given value, \(n=2\) into \((81 n^{2})^{0.75}\) to get \((81*2^{2})^{0.75}\). This simplifies to \((81*4)^{0.75} = (324)^{0.75}\).

Step 2: Simplify the expression

The next step is to evaluate \((324)^{0.75}\). The exponent of 0.75 is equivalent to taking the cube root of the number and then squaring it. So, the result will be \(\sqrt[3]{324^{2}}\)= 36.0.

Step 3: Determine the number that is closest to the result

The last step is to determine the number from the answer choices that is closest to the result. To compare these numbers, it's necessary to first convert them into decimal form if they're not already. Comparing, it's clear that 36 is closest to choice G: 76.4.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For example, if we have the expression \(a^3\), this indicates that \(a\) is multiplied by itself 3 times: \(a \times a \times a\). Exponents consist of a base number and an exponent or power number. The base is the number being multiplied, and the power tells us how many times the base is used in the multiplication.

When dealing with expressions like \((81 n^{2})^{0.75}\), you're dealing with compound exponents, where you raise a base raised to a power to another power. To simplify compound exponents, you typically multiply the powers together. Exponents can also be expressed as roots, which can make them easier to work with in operations.

Expressions

Expressions in algebra are combinations of numbers, variables, and operations. They act like phrases in the language of mathematics. An algebraic expression represents a quantity, for example, \(81 n^{2}\). This expression consists of a coefficient (81), a variable \(n\), and an exponent \(2\). By substituting values for variables, we can evaluate these expressions to obtain numerical results.

In our given exercise, substituting \(n = 2\) into \(81 n^{2}\) gives us \(81 \times 2^{2} = 324\). When expressions get more complex, such as with the inclusion of exponents, understanding how to approach each part is crucial for solving the problem efficiently.

Simplification

Simplification involves reducing a mathematical expression to its simplest form. This might involve factoring, combining like terms, or using mathematical rules to make an expression easier to work with.

In the context of our expression \((324)^{0.75}\), simplification involves recognizing the exponent \(0.75\) as \(\frac{3}{4}\), meaning you first take the cube root and then square the result. Cube root simplification \(\sqrt[3]{324}\) results in approximately 6.93, and squaring that creates the approximate final simplified value of 36.

Simplifying correctly is essential for moving efficiently through calculations and ensuring accuracy when comparing results to possible answer choices.

Problem Solving

Problem solving in algebra involves systematic steps to reach a solution, much like following a recipe in cooking. Breaking down the problem into smaller parts can make it more manageable.

For example, in our exercise, we systematically approached the problem by first substituting the variable with a given number, simplifying the expression, and then comparing it with the answer choices.

• Substitution: Identify the variable and replace it with a known value.
• Simplification: Transform the expression into its simplest form.
• Comparison: Compare the calculated result with given answer choices.

Understanding algebraic steps clearly can lead you to the correct answer without unnecessary confusion, allowing you to effectively solve similar problems independently.