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Chapter 7: Problem 49

Evaluate the expression. \(2^{6}-31\)

Short Answer

Expert verified
The evaluated expression is \(33\).

Step by step solution

01

Resolve the Power Operation

Begin by calculating \(2^{6}\). This is equal to \(64\).
02

Perform the Subtraction Operation

After the exponentiation, the expression becomes \(64 - 31\). Subtract 31 from 64 to find the final result.
03

Final Answer

Evaluating \(64 - 31\) gives a final result of \(33\).

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Key Concepts

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

Exponentiation
Exponentiation is a fundamental concept in algebra and mathematics as a whole. It refers to the process of raising a base number to the power of an exponent. For example, in the expression \(2^6\), the number 2 is the base and 6 is the exponent. This tells us how many times to multiply the base by itself.
  • In our example \(2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\).
  • This simplifies to \(64\) after performing the multiplication steps.
  • The process is efficient: instead of multiplying repeatedly, exponentiation gives a quick result for large powers.
A useful thing to remember is that any number to the power of 0 is 1, and any number to the power of 1 is the number itself.
Subtraction
Subtraction is the operation of taking one number away from another. It is one of the basic arithmetic operations and is opposite to addition. In the given expression \(64 - 31\), we perform subtraction as follows:
  • Start with the number 64, which is our result from the exponentiation.
  • Subtract 31 from 64.
  • This process involves borrowing if you're doing it on paper or a mental calculation if done quickly.
The result of \(64 - 31\) is \(33\). The concept of subtraction is straightforward: it reduces the value of the original number by the amount subtracted.
Order of Operations
The order of operations is a crucial guideline in mathematics to determine which operation to perform first in an expression. It ensures consistent and correct results when solving problems involving multiple operations.
  • Remember PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • In our problem, we handle the exponentiation \(2^6\) before the subtraction \(64 - 31\). This is due to the high precedence of exponents over addition and subtraction.
  • By following these rules, we ensure the integrity of our calculations and that everyone arrives at the same result, which in this case, is \(33\).
Keep practicing applying the order of operations, and soon it will become second nature in solving complex algebraic expressions.

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Most popular questions from this chapter

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&2 x+y=5\\\&-6 x-3 y=-15\end{aligned} $$

Decide whether the graphs of the two equations are $$ y=4 x+3 ; 2 y-8 x=-3 $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&-4 x+y=-8\\\&-12 x+3 y=-24\end{aligned} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&-7 x+7 y=7\\\&2 x-2 y=-18\end{aligned} $$

Use linear systems to determine where the \(x\) -coordinate is equal to the \(y\) -coordinate on the graph of \(2 x+3 y=25\).
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