Problem 52 - Algebra 1

Question

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 0.000094^{3} $$

Step-by-Step Solution

Answer

The result in scientific notation is \(8.8 × 10^{-11}\) and in decimal form it is \(0.000000000088\).

Step 1: Calculation

First, input the expression \(0.000094^{3}\) into the calculator.

Step 2: Convert to Scientific Notation

After obtaining the result from the calculator, convert the decimal result into scientific notation. This can be done by adjusting the decimal point and determining the number of places it moved. The scientific notation form is \(a × 10^{n}\), where 1 ≤ a < 10 and n represents the number of decimal places the decimal point moved, where it is positive if the decimal point moved to the left and negative if it moved to the right.

Step 3: Convert to Decimal Form

The decimal result from the calculator can be kept as it is to represent the decimal form.

Key Concepts

Calculator Usage

Using a calculator efficiently is fundamental for solving mathematical problems, especially when dealing with numbers as small as 0.000094. To evaluate the expression \(0.000094^{3}\), begin by turning on your calculator and ensure it is in the correct mode for power calculations. Most scientific calculators have a dedicated button for exponentiation, usually labeled as \(x^y\), \(^\wedge\), or similar.

• First, enter the base, which is 0.000094.
• Next, press the exponentiation button.
• Finally, enter the exponent value, which is 3, and press the "equals" button or the appropriate key to compute the result.

This step yields a raw decimal number on your calculator's screen, which is the result of \(0.000094^{3}\). Calculators handle these small computations with great accuracy, making them an invaluable tool for precise calculations.

Decimal Form Conversion

Once you have the raw result from the calculator, it is essential to interpret it in decimal form. The decimal form of an expression presents the number as it appears exactly—with all its digits and zeros—in a straightforward view. For \(0.000094^{3}\), after computing with the calculator, you might see a very small number such as 8.29 × 10^{-13}.To convert this back to decimal form:

• Identify the power of ten. Here, it is \(-13\), which tells us how many places the decimal point must move.
• Start at 8.29 and move the decimal point 13 places to the left. This results in 0.000000000000829.

This representation allows us to easily understand and visualize numbers on the decimal scale.

Exponentiation

Exponentiation involves raising a number to the power of another number. In our exercise, \(0.000094\) is raised to the power of 3, which means multiplying the base (0.000094) by itself three times.Mathematically, it looks like this:\[0.000094^{3} = 0.000094 \times 0.000094 \times 0.000094\]This concept is crucial for understanding how small numbers can decrease rapidly when raised to a power. Exponentiation is a fundamental operation in mathematics, extending beyond simple powers to include roots and logarithms, forming the base of algebra and many other fields.