Problem 73 - Algebra 1

Question

Simplify the expression. Then use a calculator to evaluate the expression. Round the result to the nearest tenth when appropriate. $$ \left(3.7^{3}\right)^{5} $$

Step-by-Step Solution

Answer

To obtain the final answer, calculate the numerical value of \(3.7^{15}\) and round to the nearest tenth. The exact answer depends on the calculator used, but the process to reach it has been outlined.

Step 1: Simplify the Expression Using Power of a Power Rule

The expression can be simplified as follows using the power of a power rule: \((3.7^3)^5 = 3.7^{3*5} = 3.7^{15}\).

Step 2: Evaluate the Simplified Expression

Now, using a calculator, the value of \(3.7^{15}\) can be calculated.

Step 3: Round the Result to the Nearest Tenth

The result from Step 2 will be a long decimal number. This number is rounded to the nearest tenth to obtain the final answer.

Key Concepts

Power of a Power Rule

When you come across an expression like \((a^m)^n\), you can use the power of a power rule. This rule states that you can simplify the expression by multiplying the exponents together. It’s a handy way to deal with complex expressions involving powers. For example, in the expression \((3.7^3)^5\), using the power of a power rule means multiplying the exponents 3 and 5, giving you \(3.7^{15}\).

• This rule helps simplify expressions involving nested exponents.
• It reduces complexity, making calculations easier to handle.

Remember that this rule is applicable for both numbers and variables.

Simplification

Simplification is an essential mathematical process where you reduce an expression to its most concise form without changing its value. In the context of powers and exponents, like \((3.7^3)^5\), simplification involves using the power of a power rule to streamline the expression into \(3.7^{15}\).

• Simplification can involve combining like terms.
• It often makes numbers easier to compute without losing accuracy.

The goal is to make a complex expression more manageable to work with, ensuring calculations can proceed smoothly and results are easily interpretable.

Calculator Usage

Once you've simplified an expression, using a calculator can help you evaluate it quickly and accurately. When dealing with large exponents, such as \(3.7^{15}\), manual calculations can be tedious and error-prone. Calculators can handle these computations efficiently, providing you with precise results. To compute \(3.7^{15}\) with a calculator:

• Enter the base number (3.7).
• Use the power function (often labeled as "^" or "EXP").
• Input the exponent (15).
• Hit the "equals" sign to get the result.

Calculators are valuable tools in mathematics, especially for checking work and ensuring accuracy in complex calculations.

Rounding Numbers

Rounding numbers is a mathematical technique used to reduce the number of decimal places in a number to make it simpler and more readable. After evaluating an expression like \(3.7^{15}\), you might end up with a number such as 721254941.8. Rounding this to the nearest tenth means looking at the first decimal place:

• If it’s 5 or more, round up.
• If it’s less than 5, keep the digit the same and remove all following decimals.

This process helps in making numbers more manageable, particularly when exact precision is not necessary and a general approximation will suffice.