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Chapter 2: Problem 32

Simplify. See Examples 1 through \(10\). $$ 10 \cdot 5^{3}+7 $$

Short Answer

Expert verified
The simplified expression is 1257.

Step by step solution

01

Identify the Expression

The expression to simplify is \(10 \cdot 5^3 + 7\). It consists of a multiplication component and an addition component.
02

Calculate the Power

First, calculate \(5^3\). This means multiplying 5 by itself three times: \(5 \times 5 \times 5 = 125\).
03

Multiply the Result

Now, take the result from Step 2, which is 125, and multiply it by 10, as indicated: \(10 \cdot 125 = 1250\).
04

Add the Remaining Term

Finally, add the last term, 7, to the result from Step 3: \(1250 + 7 = 1257\).
05

Conclusion: Simplified Expression

The simplified expression is \(1257\). This is the final result after completing all calculations.

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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 one number, known as the base, to the power of another number, known as the exponent. The expression \(5^3\) in our example indicates that 5 should be multiplied by itself a total of three times. Understanding this process:
  • The base is the number to be multiplied.
  • The exponent tells us how many times to use the base as a factor in the multiplication.
Given this, we calculate \(5^3\) as follows: 1. Multiply 5 by 5, which equals 25.2. Then, multiply the result, 25, by 5 again, equaling 125.Thus, \(5^3 = 125\). This is a crucial initial step that sets the stage for further operations in the expression.
Multiplication
Multiplication is one of the basic arithmetic operations that involves scaling one number by another. In our expression, we now multiply the result of the exponentiation, 125, by 10. This step follows the order of operations, ensuring we deal with multiplication before addition.Here鈥檚 how it works:
  • Take the product from exponentiation: 125.
  • Multiply 125 by 10, the coefficient of the term.
This calculation yields a result of: \( 10 \times 125 = 1250 \).Multiplication transforms the prior result into a form that can then be easily combined with additional terms in the expression.
Addition
Addition is the final step we take to complete our expression simplification. After completing both the exponentiation and multiplication, we have a remaining term to add to our current total. Addition allows us to combine quantities.Key details to remember:
  • Start with the result of previous operations, which is 1250.
  • Add the last term in the expression to this result.
Performing this final calculation:\( 1250 + 7 = 1257 \).Here, addition combines all previous results into a single, simplified expression. This final value of 1257 represents the complete, simplified form of our original mathematical problem.

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