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Chapter 4: Problem 35

Evaluate each exponential expression. $$ 2^{6} $$

Short Answer

Expert verified
The value of \(2^{6}\) is 64.

Step by step solution

01

- Understand the Expression

The given expression is an exponential notation: \(2^{6}\). This means 2 is the base and 6 is the exponent.
02

- Expand the Exponential Expression

Write the expression in expanded form: \(2^{6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2\).
03

- Perform the Multiplication

Now, multiply the terms step-by-step: 1. \(2 \times 2 = 4\) 2. \(4 \times 2 = 8\) 3. \(8 \times 2 = 16\) 4. \(16 \times 2 = 32\) 5. \(32 \times 2 = 64\)
04

- Conclusion

After performing the multiplication, the value of the exponential expression \(2^{6}\) is 64.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

base and exponent
In the expression \(2^{6}\), we have two main components: the base and the exponent. The base is the number that is being multiplied, which, in this case, is 2. The exponent tells us how many times to multiply the base by itself, which is 6 times for this exercise.
To better visualize this:
  • The base is 2.
  • The exponent is 6.
Together, they tell us to multiply 2 by itself a total of 6 times. Understanding these components is crucial for working with exponential expressions.
expanded form
Writing an exponential expression in expanded form helps us to see all the multiplications involved. For \(2^{6}\), we rewrite it as:
\(2^{6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2\)
This expanded form shows that we need to multiply six 2's together. Breaking it down step-by-step makes it more manageable, especially for larger exponents. Each step represents multiplying the base, 2, by the result of the previous multiplication. Using the expanded form is a great way to simplify and understand complex exponential expressions.
multiplication steps
Now we will calculate the value of \(2^{6}\) by multiplying step-by-step. Here are the detailed steps:
  • Start with the first two 2's: \(2 \times 2 = 4\).
  • Next, multiply the result by the next 2: \(4 \times 2 = 8\).
  • Continue: \(8 \times 2 = 16\).
  • Then: \(16 \times 2 = 32\).
  • Finally: \(32 \times 2 = 64\).
After following through each step, we find that \(2^{6} = 64\). Doing the multiplication gradually helps to avoid mistakes and ensures you get the correct result.

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