Problem 49 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(2^{7}\)

Step-by-Step Solution

Answer

The value of \(2^7\) is 128.

Step 1: Understand the Power Notation

The expression \(2^7\) is read as '2 raised to the power of 7'. This means that the number 2 is multiplied by itself a total of 7 times.

Step 2: Expand the Expression

To calculate \(2^7\), expand the expression as follows: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\).

Step 3: Perform Multiplication in Steps

Start multiplying the numbers in pairs or groups:- First, calculate \(2 \times 2 = 4\).- Next, multiply the result by the next 2: \(4 \times 2 = 8\).- Continue with \(8 \times 2 = 16\).- \(16 \times 2 = 32\).- \(32 \times 2 = 64\).- Finally, \(64 \times 2 = 128\).

Step 4: Verify with a Calculator

Use a calculator to verify the result of \(2^7\) by entering the base (2) and exponent (7) into the calculator. The calculator should confirm the result as 128.

Key Concepts

Power Notation

Power notation is a mathematical shorthand used to simplify the representation of repeated multiplication of the same number. If you see something like "\(2^7\)", it indicates that 2 is multiplied by itself 7 times. In simple terms, it's saying "2 times 2 times 2," and so on, until you've done it seven times. This is particularly useful for dealing with large calculations where a number is involved many times.

In power notation, the base is the number being multiplied, and the exponent tells you how many times the base multiplies itself. Here:

• The base is 2.
• The exponent is 7.

The expanded form is simply a way of writing down all the multiplications, such as \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\). Understanding and recognizing this pattern makes complex arithmetic more manageable.

Multiplication

Multiplication of repeated numbers, especially in the context of powers, is incremental and simple when broken down into steps. Once you comprehend the power notation, the next phase is actually performing the multiplication.

When handling exponents, multiply in manageable steps:

• First, pair off or group the numbers to make multiplication easier. For \(2^7\), start with \(2 \times 2 = 4\).
• Then, multiply this result by the next 2: \(4 \times 2 = 8\).
• Keep going until all the multiplications are done, such as \(8 \times 2 = 16\), \(16 \times 2 = 32\), \(32 \times 2 = 64\), and finally \(64 \times 2 = 128\).

This comprehensive, step-by-step approach is often easier for calculations by hand and ensures accurate results. As you get comfortable, you may group larger sets or even memorize certain powers for quicker usage.

Calculator Verification

Calculators are handy for checking complex calculations or minimizing errors, especially with powers. To verify your calculations of powers, such as \(2^7\), use a calculator as follows:

• Find the power function, often represented by a "^" button or through the use of parentheses for inputting powers.
• Input the base number, 2 in this case.
• Enter the power: 7.
• Press equals or execute the command to see the result.

The calculator should confirm your manual calculations, showing 128.

Using calculators not only helps verify results but also builds confidence in manual calculations, and helps ensure the understanding is solid before moving to more complicated problems.