Problem 69 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation. Write the result in scientific notation. (Lesson 8.5). $$ (2.5 \times 10)^{-2} $$

Step-by-Step Solution

Answer

The result of \( (2.5 \times 10)^{-2} \) in scientific notation is \( 1.6 \times 10^{-3}\)

Step 1: Perform the operation

First, perform the operation for the base, which is multiplication. Multiply 2.5 by 10, you get 25. So, (2.5 x 10)^{-2} becomes (25)^{-2}.

Step 2: Apply the exponent

A nonzero number raised to the power -n is the same as 1 divided by that number to the power n. Hence, (25)^{-2} equals 1 divided by 25 squared, or \( \frac{1}{25^{2}} \).

Step 3: Calculate the denominator

Calculate the denominator part to get the final answer. The square of 25 is 625. So, \( \frac{1}{25^{2}} \) is \( \frac{1}{625} \)

Step 4: Convert to Scientific Notation

To express \( \frac{1}{625} \) in scientific notation, it can be converted to 0.0016. With the decimal moved to just after the first nonzero number, you get 1.6 x 10. Since the actual decimal point was moved three places to the right, the 10 gets an exponent of -3. So \( \frac{1}{625} \) in scientific notation is \(1.6 \times 10^{-3}\)

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. When you see a number with an exponent, it tells you to use that number as a factor. For instance, in the expression \(2^3\), it means you multiply 2 by itself three times: \(2 \times 2 \times 2\).

Exponents can simplify expressions and make calculations easier. When working with exponents, remember these key rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)


• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^m = a^m \times b^m\)

Using these rules, you can handle more complex problems involving repeated multiplication.

Working with Negative Exponents

Negative exponents might seem a bit tricky at first, but they follow a simple rule. A negative exponent means you take the reciprocal of the base and then apply the positive exponent. For example, \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).

This rule helps to understand expressions like \( (2.5 \times 10)^{-2} \). Instead of directly calculating each number, think of flipping and dealing with positive exponents. Let's consider \(25^{-2}\). Here, it's 1 divided by \(25^2\), which equals \(\frac{1}{625}\).

Negative exponents might seem like dividing. In reality, they simplify expressions and are particularly helpful when converting to scientific notation. This process is crucial for expressing fractions in a more manageable form with a power of ten.

Multiplication and Scientific Notation

When multiplying numbers in scientific notation, the key is to handle the coefficients and the powers of ten separately. Numbers in scientific notation are written as \(a \times 10^b\), where \(a\) is a number between 1 and 10, and \(b\) is an integer.

Consider multiplying \(2.5 \times 10\) from the problem, which gives 25. In scientific notation:

• First multiply the base numbers. Here, \(2.5 \times 10 = 25\).
• Convert the result, if needed, into scientific notation: 25 becomes \(2.5 \times 10^1\).

Now, if you're working with scientific notation and negative exponents, like \(\frac{1}{625}\), express the number 0.0016 as \(1.6 \times 10^{-3}\).

Scientific notation helps simplify working with small numbers and makes them easier to understand by focusing on the significant digit and the scale (positive or negative exponent) separately.