Question
4 The expression \( 6^{3} \) has a value of 216 . Write three more expressions with exponents that also have a value of 216.
Ask by Logan Pierce. in the United States
Feb 25,2025
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Answer
Here are three expressions with exponents that equal 216:
1. \( 2^3 \times 3^3 \)
2. \( 6^3 \)
3. \( 216^1 \)
Solution
To find three more expressions with exponents that equal 216, we can express 216 in different ways using bases and exponents.
1. **Using a different base**: We can express 216 as \( 6^3 \) or \( 2^n \) for some \( n \).
2. **Finding the prime factorization**: The prime factorization of 216 is:
\[
216 = 2^3 \times 3^3
\]
This means we can use combinations of these prime factors to create expressions.
Now, let's create three different expressions:
1. **Using base 2**:
\[
216 = (2^3) \times (3^3) = 6^3 \implies 216 = (2 \times 3)^3 = 6^3
\]
2. **Using base 3**:
\[
216 = 3^3 \times 2^3 = (3 \times 2)^3 = 6^3
\]
3. **Using a combination of bases**:
\[
216 = 2^3 \times 3^3 = (2 \times 3)^{3} = 6^3
\]
Now, let's express 216 in terms of different bases and exponents:
1. \( 2^3 \times 3^3 \)
2. \( 6^3 \)
3. \( 216^1 \)
Thus, the three expressions with exponents that also have a value of 216 are:
1. \( 2^3 \times 3^3 \)
2. \( 6^3 \)
3. \( 216^1 \)
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Did you know that \( 216 \) is also the result of \( 6^3 \) because it’s the cube of 6? Besides that, you can express \( 216 \) in different ways using exponents, such as \( 2^3 \times 3^3 \) or \( 12^2 \) since \( 12 \times 12 = 144 \) which requires further math to reach 216. Let’s take a fun peek into \( 216 \) in terms of bases and exponents: you can write it as \( 3^3 \times 4^3 \) since \( (3 \times 4) = 12 \) and \( 12^2 = 144 \). Another expressive twist is using \( 216 \) as \( 6^{\frac{3}{1}} \), enhancing your foundational understanding of exponents through creativity!
