Asked by bluesanta
3sqrt64/27^⅓ - simplify
can you do step by step pls
can you do step by step pls
Answers
Answered by
GPT-4o mini
Sure! Let's simplify the expression step by step:
\[
\frac{3\sqrt{64}}{27^{1/3}}
\]
### Step 1: Simplify \(\sqrt{64}\)
The square root of 64 is 8, because \(8 \times 8 = 64\).
Now, we can rewrite the expression:
\[
\frac{3 \cdot 8}{27^{1/3}}
\]
### Step 2: Multiply the numerator
Now, multiply \(3\) by \(8\):
\[
3 \cdot 8 = 24
\]
So, the expression becomes:
\[
\frac{24}{27^{1/3}}
\]
### Step 3: Simplify \(27^{1/3}\)
Now we need to simplify \(27^{1/3}\). The cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\).
Thus, we can rewrite the expression:
\[
\frac{24}{3}
\]
### Step 4: Simplify the fraction
Now, we simplify the fraction \(\frac{24}{3}\):
\[
24 \div 3 = 8
\]
### Final Answer
Thus, the simplified form of the original expression \(\frac{3\sqrt{64}}{27^{1/3}}\) is:
\[
8
\]
\[
\frac{3\sqrt{64}}{27^{1/3}}
\]
### Step 1: Simplify \(\sqrt{64}\)
The square root of 64 is 8, because \(8 \times 8 = 64\).
Now, we can rewrite the expression:
\[
\frac{3 \cdot 8}{27^{1/3}}
\]
### Step 2: Multiply the numerator
Now, multiply \(3\) by \(8\):
\[
3 \cdot 8 = 24
\]
So, the expression becomes:
\[
\frac{24}{27^{1/3}}
\]
### Step 3: Simplify \(27^{1/3}\)
Now we need to simplify \(27^{1/3}\). The cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\).
Thus, we can rewrite the expression:
\[
\frac{24}{3}
\]
### Step 4: Simplify the fraction
Now, we simplify the fraction \(\frac{24}{3}\):
\[
24 \div 3 = 8
\]
### Final Answer
Thus, the simplified form of the original expression \(\frac{3\sqrt{64}}{27^{1/3}}\) is:
\[
8
\]
Answered by
GPT-4o mini
Sure! Let's simplify the expression \(\frac{3\sqrt{64}}{27^{1/3}}\) using smaller numbers when possible.
### Step 1: Simplify \(\sqrt{64}\)
\(\sqrt{64} = 8\)
### Step 2: Rewrite the expression
Now, substituting that back in:
\[
\frac{3 \cdot 8}{27^{1/3}}
\]
### Step 3: Multiply the numerator
Now, multiply:
\[
3 \cdot 8 = 24
\]
So the expression becomes:
\[
\frac{24}{27^{1/3}}
\]
### Step 4: Simplify \(27^{1/3}\)
Next, we simplify \(27^{1/3}\):
\[
27^{1/3} = 3
\]
### Step 5: Rewrite the expression
So we can rewrite the expression now as:
\[
\frac{24}{3}
\]
### Step 6: Simplify the fraction
Now we simplify \(\frac{24}{3}\):
\[
24 \div 3 = 8
\]
### Final Answer
The simplified form of \(\frac{3\sqrt{64}}{27^{1/3}}\) remains:
\[
8
\]
If you mean to present smaller numbers or different simplifications (like factoring, or using smaller values in other contexts), please clarify further!
### Step 1: Simplify \(\sqrt{64}\)
\(\sqrt{64} = 8\)
### Step 2: Rewrite the expression
Now, substituting that back in:
\[
\frac{3 \cdot 8}{27^{1/3}}
\]
### Step 3: Multiply the numerator
Now, multiply:
\[
3 \cdot 8 = 24
\]
So the expression becomes:
\[
\frac{24}{27^{1/3}}
\]
### Step 4: Simplify \(27^{1/3}\)
Next, we simplify \(27^{1/3}\):
\[
27^{1/3} = 3
\]
### Step 5: Rewrite the expression
So we can rewrite the expression now as:
\[
\frac{24}{3}
\]
### Step 6: Simplify the fraction
Now we simplify \(\frac{24}{3}\):
\[
24 \div 3 = 8
\]
### Final Answer
The simplified form of \(\frac{3\sqrt{64}}{27^{1/3}}\) remains:
\[
8
\]
If you mean to present smaller numbers or different simplifications (like factoring, or using smaller values in other contexts), please clarify further!
Answered by
GPT-4o mini
Let's simplify the expression \((4\sqrt{8})^{-24} \times (\sqrt{8})^{16}\) step by step.
### Step 1: Simplify \(\sqrt{8}\)
The square root of 8 can be simplified:
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\]
### Step 2: Substitute \(\sqrt{8}\) into the expression
Now we can substitute \(\sqrt{8}\) with \(2\sqrt{2}\):
\[
(4\sqrt{8})^{-24} \times (\sqrt{8})^{16} = (4(2\sqrt{2}))^{-24} \times (2\sqrt{2})^{16}
\]
### Step 3: Simplify \(4\sqrt{8}\)
Now we simplify \(4(2\sqrt{2})\):
\[
4\sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2}
\]
Now we can rewrite the expression:
\[
(8\sqrt{2})^{-24} \times (2\sqrt{2})^{16}
\]
### Step 4: Simplifying \((8\sqrt{2})^{-24}\)
Now let's handle the \((8\sqrt{2})^{-24}\):
\[
(8\sqrt{2})^{-24} = 8^{-24}(\sqrt{2})^{-24} = 8^{-24} \cdot (2^{1/2})^{-24} = 8^{-24} \cdot 2^{-12}
\]
Since \(8 = 2^3\), we can further simplify:
\[
(2^3)^{-24} = 2^{-72}
\]
So:
\[
(8\sqrt{2})^{-24} = 2^{-72} \cdot 2^{-12} = 2^{-72 - 12} = 2^{-84}
\]
### Step 5: Simplifying \((2\sqrt{2})^{16}\)
Now let's simplify \((2\sqrt{2})^{16}\):
\[
(2\sqrt{2})^{16} = (2(2^{1/2}))^{16} = (2^{1 + 1/2})^{16} = (2^{3/2})^{16} = 2^{(3/2) \cdot 16} = 2^{24}
\]
### Step 6: Combine the results
Now we have:
\[
(4\sqrt{8})^{-24} \times (\sqrt{8})^{16} = 2^{-84} \times 2^{24}
\]
Combine the exponents:
\[
2^{-84 + 24} = 2^{-60}
\]
### Final Answer
So the simplified expression is:
\[
\frac{1}{2^{60}} \quad \text{or} \quad 2^{-60}
\]
### Step 1: Simplify \(\sqrt{8}\)
The square root of 8 can be simplified:
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\]
### Step 2: Substitute \(\sqrt{8}\) into the expression
Now we can substitute \(\sqrt{8}\) with \(2\sqrt{2}\):
\[
(4\sqrt{8})^{-24} \times (\sqrt{8})^{16} = (4(2\sqrt{2}))^{-24} \times (2\sqrt{2})^{16}
\]
### Step 3: Simplify \(4\sqrt{8}\)
Now we simplify \(4(2\sqrt{2})\):
\[
4\sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2}
\]
Now we can rewrite the expression:
\[
(8\sqrt{2})^{-24} \times (2\sqrt{2})^{16}
\]
### Step 4: Simplifying \((8\sqrt{2})^{-24}\)
Now let's handle the \((8\sqrt{2})^{-24}\):
\[
(8\sqrt{2})^{-24} = 8^{-24}(\sqrt{2})^{-24} = 8^{-24} \cdot (2^{1/2})^{-24} = 8^{-24} \cdot 2^{-12}
\]
Since \(8 = 2^3\), we can further simplify:
\[
(2^3)^{-24} = 2^{-72}
\]
So:
\[
(8\sqrt{2})^{-24} = 2^{-72} \cdot 2^{-12} = 2^{-72 - 12} = 2^{-84}
\]
### Step 5: Simplifying \((2\sqrt{2})^{16}\)
Now let's simplify \((2\sqrt{2})^{16}\):
\[
(2\sqrt{2})^{16} = (2(2^{1/2}))^{16} = (2^{1 + 1/2})^{16} = (2^{3/2})^{16} = 2^{(3/2) \cdot 16} = 2^{24}
\]
### Step 6: Combine the results
Now we have:
\[
(4\sqrt{8})^{-24} \times (\sqrt{8})^{16} = 2^{-84} \times 2^{24}
\]
Combine the exponents:
\[
2^{-84 + 24} = 2^{-60}
\]
### Final Answer
So the simplified expression is:
\[
\frac{1}{2^{60}} \quad \text{or} \quad 2^{-60}
\]
Answered by
GPT-4o mini
Let's simplify the expression \(\frac{7^{-4} \times 7^{3}}{\sqrt{49}}\) step by step.
### Step 1: Simplify \(\sqrt{49}\)
The square root of 49 is:
\[
\sqrt{49} = 7
\]
### Step 2: Rewrite the expression
Now, substitute \(\sqrt{49}\) with \(7\):
\[
\frac{7^{-4} \times 7^{3}}{7}
\]
### Step 3: Simplify the numerator
Using the property of exponents, \(a^m \times a^n = a^{m+n}\):
\[
7^{-4} \times 7^{3} = 7^{-4 + 3} = 7^{-1}
\]
### Step 4: Rewrite the fraction
Now the expression becomes:
\[
\frac{7^{-1}}{7}
\]
### Step 5: Simplify the fraction
Again, using the property of exponents, we can express the denominator \(7\) as \(7^{1}\):
\[
\frac{7^{-1}}{7^{1}} = 7^{-1 - 1} = 7^{-2}
\]
### Step 6: Final Answer
Thus, the simplified form of the original expression \(\frac{7^{-4} \times 7^{3}}{\sqrt{49}}\) is:
\[
7^{-2} \quad \text{or} \quad \frac{1}{7^2} = \frac{1}{49}
\]
### Step 1: Simplify \(\sqrt{49}\)
The square root of 49 is:
\[
\sqrt{49} = 7
\]
### Step 2: Rewrite the expression
Now, substitute \(\sqrt{49}\) with \(7\):
\[
\frac{7^{-4} \times 7^{3}}{7}
\]
### Step 3: Simplify the numerator
Using the property of exponents, \(a^m \times a^n = a^{m+n}\):
\[
7^{-4} \times 7^{3} = 7^{-4 + 3} = 7^{-1}
\]
### Step 4: Rewrite the fraction
Now the expression becomes:
\[
\frac{7^{-1}}{7}
\]
### Step 5: Simplify the fraction
Again, using the property of exponents, we can express the denominator \(7\) as \(7^{1}\):
\[
\frac{7^{-1}}{7^{1}} = 7^{-1 - 1} = 7^{-2}
\]
### Step 6: Final Answer
Thus, the simplified form of the original expression \(\frac{7^{-4} \times 7^{3}}{\sqrt{49}}\) is:
\[
7^{-2} \quad \text{or} \quad \frac{1}{7^2} = \frac{1}{49}
\]
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