Problem 41 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ 2^{-5} $$

Step-by-Step Solution

Answer

The solution for the expression \(2^{-5}\) rounded to the nearest ten thousandth is 0.03125.

Step 1: Understand the Negative Exponent

A negative exponent means to take the reciprocal of the base and then raise it to the power of the absolute value of the exponent. So, \(2^{-5}\) is actually equivalent to \(1/(2^5)\).

Step 2: Calculate the Power of Positive Exponent

Calculate \(2^5\), which equals to 32.

Step 3: Calculate the Reciprocal of the Result

Calculate the reciprocal of 32, we do the calculation \(1/32\), which gives the decimal 0.03125.

Step 4: Round the Answer

Round the result 0.03125 to the nearest ten thousandth which results the same: 0.03125.

Key Concepts

Reciprocal

Reciprocal is a math term that refers to the flip of a fraction. Imagine you have a fraction like \( \frac{a}{b} \). Its reciprocal would be \( \frac{b}{a} \). In simple terms, to find the reciprocal, you just switch the positions of the numerator and the denominator. For whole numbers, like 5, its reciprocal is \( \frac{1}{5} \) because every whole number has an implicit denominator of 1.

Reciprocals are very useful when dealing with negative exponents. A negative exponent indicates that we take the reciprocal of the base. For instance, in the problem with \( 2^{-5} \), we interpret it as the reciprocal of \( 2^5 \), which is \( \frac{1}{32} \). This shifting of position highlights the power of reciprocals in simplifying expressions with negative exponents.

Exponentiation

Exponentiation is a way of expressing repeated multiplication of a number by itself. An exponent (or power) tells you how many times to use the number in a multiplication. For example, \( 2^5 \) means we multiply 2 by itself a total of 5 times, resulting in 32. This sequence looks like:

• 2 x 2 = 4
• 4 x 2 = 8
• 8 x 2 = 16
• 16 x 2 = 32

So, the answer for \( 2^5 \) is 32.

When exponents are negative, like in \( 2^{-5} \), it indicates a division instead of multiplication. As mentioned, this is effectively taking the reciprocal of \( 2^5 \). Hence, \( 2^{-5} \) turns into \( \frac{1}{32} \), emphasizing how negative exponents transform the operation from multiplication to division.

Rounding Numbers

Rounding numbers refers to adjusting the digits of a number to make it simpler, while keeping its value close to what it was. The goal is to make numbers easier to work with.

For example, the exercise asks us to round to the nearest ten thousandth. Let's break that down:

• The first digit after the decimal point is the tenths place.
• The second digit is the hundredths place.
• The third is the thousandths place.
• The fourth is the ten thousandths place.

So, for 0.03125, the digit in the ten thousandths place is 2. Since the number following it (5) is 5 or higher, you round the 2 up to a 3, resulting in the value staying as 0.03125.

Rounding is an essential skill, making estimates easy and numbers more intuitive, especially when high precision isn't necessary.