Problem 46 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ \left(3^{-3}\right)^{2} $$

Step-by-Step Solution

Answer

The result of the expression, rounded to the nearest ten thousandth, is 0.0014.

Step 1: Calculate the exponent

Calculate the value of \(3^{-3}\). In general, any number (let us denote it as a) to the power of -n (a negative number) is equivalent to \(1/a^n\). So, \(3^{-3}\) is equivalent to \(1/3^3 = 1/27\).

Step 2: Squaring the result

Next, we need to square the result from step 1, which is \(1/27\). Squaring means multiplying the number by itself, so we need to calculate \((1/27)^2\), which equals \(1/729\).\}

Step 3: Convert to decimal

To complete the exercise, the result needs to be in decimal form. So, we need to convert \(1/729\) to a decimal. This can be done using a calculator, which gives approximately 0.00137174211288.

Step 4: Round to the nearest ten thousandth

The last step is to round the decimal number from Step 3 to the nearest ten thousandth. Rounding 0.00137174211288 to the nearest ten thousandth gives 0.0014.

Key Concepts

Negative exponents

Negative exponents can be a little tricky at first, but they are not too difficult to grasp with a bit of practice. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. In simpler terms, if you have a number like \(a^{-n}\), it becomes \(\frac{1}{a^n}\).

Let's take the number \(3^{-3}\):

• First, change the negative exponent to a positive one by flipping the base: \(3^{-3} = \frac{1}{3^3}\).
• Then, calculate \(3^3\), which is 27.
• So, \(3^{-3} = \frac{1}{27}\).

Understanding negative exponents can help you better navigate problems involving reciprocals. This skill is quite useful in algebra and beyond.

Squaring a number

Squaring a number is a fundamental concept that means multiplying the number by itself. If you have \((x)^2\), you are simply computing \(x \times x\). It’s straightforward, but very important.

For example, consider the expression \(\left(3^{-3}\right)^{2}\). When you squared \(3^{-3}\), you had already converted it to a fraction \(\frac{1}{27}\). Now, squaring \(\frac{1}{27}\) means:

• Multiply \(\frac{1}{27} \times \frac{1}{27}\).
• This results in \(\frac{1}{729}\) because \(27 \times 27 = 729\).

Squaring is a building block for many math concepts like quadratic equations and geometric problems. Mastering it will help you with countless other math challenges.

Decimal conversion

Decimal conversion is the process of converting a fraction into its decimal form. This is a key step in making numbers more understandable, especially when you need to apply them in practical scenarios. Converting fractions to decimals can be easily done using a calculator.

Let's see how this applies to the fraction \(\frac{1}{729}\):

• Enter the fraction \(\frac{1}{729}\) into a calculator.
• You will get an approximate decimal outcome: 0.00137174211288.

Decimals are often preferred in scientific and financial calculations for their ease of use and ability to represent numbers in a linear form.

Rounding numbers

Rounding numbers is a vital skill that helps to simplify complex calculations and make results easier to interpret. The process involves finding the closest value at a specified decimal place or whole number. It's particularly useful for reporting measurements or making estimations.

To round a number like 0.00137174211288 to the nearest ten thousandth, follow these steps:

• Notice that you need to round to four decimal places.
• Look at the fifth digit (the place after your desired precision), which is 7.
• Since 7 is greater than 5, you round up the fourth place number from 3 to 4.
• The rounded number is 0.0014.

Rounding helps you summarize data and make it more accessible without losing significant accuracy.