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Chapter 1: Problem 72

Simplify the given expression. $$ 4^{3}-(6+5) $$

Short Answer

Expert verified
53

Step by step solution

01

Simplify the exponent

Calculate the value of \(4^3\). This means multiplying 4 by itself three times: \(4 \times 4 \times 4 = 64\).
02

Simplify inside the parentheses

Calculate \((6 + 5)\). Add 6 and 5 to get 11.
03

Substitute and Subtract

Now replace the cubic expression and the parenthetical expression in the original equation: \(64 - 11\). Subtract 11 from 64 to get 53.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, which is the exponent. In the expression \(4^3\), 4 is the base, and 3 is the exponent. To calculate \(4^3\), we need to multiply the base, 4, by itself a total of three times.
So, it would look like:
\[4 \times 4 \times 4 \]
Breaking it down:
  • First, multiply the first two 4s: \(4 \times 4 = 16\).
  • Then multiply the result with the last 4: \(16 \times 4 = 64\).
This final result, 64, is the value of \(4^3\).
Exponentiation is crucial for simplifying expressions accurately, as it dictates how many times the base is multiplied by itself.
Parentheses
Parentheses are symbols used in mathematics to group parts of an expression. They are essential to indicating which operations need to be performed first. According to the order of operations (PEMDAS/BODMAS), calculations inside parentheses take precedence over others like multiplication or subtraction.
In our example, we have the expression \((6 + 5)\).
Inside these parentheses, we have an addition operation:
  • Add 6 and 5 to get 11.
This means inside the parentheses, the expression simplifies to the value of 11.
Always remember that solving expressions inside parentheses first helps keep calculations organized and accurate.
Subtraction
Subtraction is the operation used to represent the taking away of the value of one number from another. It is denoted by the minus sign (-). In our given expression after simplification, we substitute the results from the previous steps. It becomes \(64 - 11\).
Let's solve it step by step:
  • Start with 64 as the minuend, the number you start with.
  • The subtrahend in this case is 11, the number to be subtracted.
  • Perform the subtraction: \(64 - 11 = 53\).
This result shows how subtraction helps in finding the difference between numbers, which is vital in simplifying arithmetic expressions.

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