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Chapter 2: Problem 2

Evaluate the given expression. Do not use a calculator. $$ 4^{3}-3^{4} $$

Short Answer

Expert verified
The evaluated expression \(4^3 - 3^4 = 64 - 81 = -17\).

Step by step solution

01

Calculate \(4^3\)

To find the cube of 4, we need to multiply 4 by itself three times: \(4^3 = 4 \times 4 \times 4 = 64\).
02

Calculate \(3^4\)

To find the fourth power of 3, we need to multiply 3 by itself four times: \(3^4 = 3 \times 3 \times 3 \times 3= 81\).
03

Subtract the two terms

Now that we have found both \(4^3\) and \(3^4\), we can subtract them to find the final answer: \(4^3 - 3^4 = 64 - 81 = -17\). The evaluated expression without using a calculator is -17.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Powers of Numbers
The concept of powers, or exponentiation, is fundamental in mathematics. It refers to multiplying a number by itself a certain number of times. Exponents are written as tiny numbers (called superscripts) above the base number. For example, in the expression \(4^3\), the 3 is the exponent and indicates how many times the base number 4 is used in multiplication.
  • **Basic idea**: An exponent, or power, tells us how many times to use the base in a multiplication.
  • **Example**: \(4^3 = 4 \times 4 \times 4\). Here, 4 is multiplied by itself three times, resulting in 64.
  • **Notation**: The expression can be pronounced as "four to the power of three" or "four cubed."
Learning to understand and calculate powers is crucial for solving more complex mathematical problems and simplifying expressions.
Arithmetic Operations
Arithmetic operations are the basic calculations you perform daily. They consist of addition, subtraction, multiplication, and division. In the expression \(4^3 - 3^4\), we perform these operations involving powers.
Once you calculate the powers, you need to follow proper arithmetic operations to solve the expression:
  • **Multiplication as repeated addition**: Calculating powers involves multiplying numbers repeatedly, which is a foundational arithmetic operation.

  • **Subtraction**: After calculating \(4^3\) and \(3^4\), subtraction is used as the final operation to evaluate the expression.
Thus, using arithmetic operations accurately and methodically is necessary to obtain the correct result in complex expressions.
Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value. They can be simple or complex and are used to express mathematical ideas.
In the original problem \(4^3 - 3^4\), the expression consists of two exponentiated terms followed by a subtraction operator. Here's how you understand and tackle such expressions:
  • **Expression components**: Identify the different parts, such as numbers or variables, and recognize what operations are being used.

  • **Order of Operations**: Always calculate powers first before other operations. It's part of the PEMDAS/BODMAS rules standing for Parentheses, Exponents (or Orders), Multiplication and Division, Addition and Subtraction.
  • **Combining results**: After finding powers, combine them using the appropriate arithmetic operation, like subtraction in this case.
Mastering mathematical expressions is essential for solving equations and developing critical problem-solving skills.

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Most popular questions from this chapter

Find all real numbers \(x\) that satisfy the indicated equation. $$ x^{4}-3 x^{2}=10 $$

Sketch the graph of the given function \(f\) on the interval [-1.3,1.3] $$ f(x)=-3 x^{3}+4 $$

Suppose \(x\) is a real number and \(m, n,\) and \(p\) are positive integers. Explain why $$ x^{m+n+p}=x^{m} x^{n} x^{p} $$

Using the result that \(\sqrt{2}\) is irrational, explain why \(2^{1 / 6}\) is irrational.

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{1 / 11} $$
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