Problem 51 - Algebra 1

Question

Use a calculator to evaluate the expression. Round your answer to the nearest hundred thousandth. $$55 \cdot 5^{-6}$$

Step-by-Step Solution

Answer

The rounded result of the expression \(55 \cdot 5^{-6}\) to the nearest hundred thousandth is 0.0000176.

Step 1: Understanding the exponent

We need to understand that \(5^{-6}\) means 1 divided by \(5^{6}\). In general, any number to the power of -n equals 1 divided by that number to the power of n.

Step 2: Calculate \(5^{-6}\)

Now we calculate \(5^{-6}\) using a calculator. You should get 0.00000032.

Step 3: Multiply by 55

Multiply the result from Step 2 by 55. Use a calculator to do this. 0.00000032 multiplied by 55 equals 0.0000176.

Step 4: Round to the nearest hundred thousandth

The result in Step 3 is already in the hundred thousandth place, and no rounding is necessary. Therefore, the final answer is as obtained in Step 3, which is 0.0000176.

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers. This notation makes communication of such numbers simple and reduces confusion. In scientific notation, numbers are written as a product of a number between 1 and 10 and a power of ten.

When dealing with very small numbers, as in the exercise, scientific notation provides an efficient way to represent them. For instance, the number 0.0000176 can be expressed in scientific notation as \(1.76 \times 10^{-5}\). This format can make calculations easier and helps avoid errors in arithmetic operations.

Additionally, scientific notation is not just a mathematical formality but an essential tool in various fields like engineering and science where dealing with extremely large or small numbers is common. It's also very useful in computational settings, including calculators, ensuring operations remain precise and manageable.

Calculators in Mathematics

Calculators play a pivotal role in modern mathematics by providing quick solutions to complex calculations. They are especially useful when you are dealing with exponents, as seen in the exercise with the expression \(5^{-6}\).

Calculators help by automatically handling the intricacies of exponentiation and multiplication, allowing for accurate results with minimal chance of manual error. When entering expressions like \(5^{-6}\) into a calculator, you don't have to worry about the long division steps required to calculate the power; the calculator processes this instantly.

Moreover, calculators can directly perform multiple operations in sequence. In our example, after finding the value for \(5^{-6}\), you can multiply that result by 55 right away, simplifying the arithmetic process overall. This provides clarity and confidence in classroom settings, letting students focus more on understanding the underlying concepts rather than getting bogged down in arithmetic details.

Rounding Numbers

Rounding numbers is a fundamental mathematical skill that ensures results are both convenient and practical without compromising much on accuracy.

In mathematics, rounding involves adjusting numbers to reduce their digits while keeping them close to their original value. This is particularly helpful when presenting results, such as rounding 0.0000176 in the exercise to the nearest hundred thousandth.

When rounding to the nearest hundred thousandth, you focus on the digit in the hundred thousandth place, which is the sixth digit to the right of the decimal in this context. If the following digit is 5 or greater, you round up; if it is less than 5, you leave the digit unchanged. In this exercise, the number 0.0000176 doesn't require rounding since the essential digit remains unchanged. Rounding ensures that numbers are simple to read and use, especially when precision beyond a point is unnecessary.