What is 6 to the 4th power multiplied by 6 to the 4th power as a single power
The Correct Answer and Explanation is:
To solve the expression 64×646^4 \times 6^464×64 as a single power, you can apply the product of powers property in exponents, which states:am×an=am+na^m \times a^n = a^{m + n}am×an=am+n
In your case, both exponents are the same, so the expression becomes:64×64=64+4=686^4 \times 6^4 = 6^{4 + 4} = 6^864×64=64+4=68
Now, let’s break down why this works:
- Product of Powers Property: When you multiply terms with the same base (here, base 6), you add the exponents together. This property simplifies the expression to just one power of 6, with the exponent being the sum of the exponents.
- Simplification: Instead of calculating 646^464 and then multiplying the results, this method allows you to combine the terms directly into a single power, which is computationally more efficient and easier to handle.
- Example Verification:
- To compute 646^464, first calculate 6×6=366 \times 6 = 366×6=36, then multiply 36×6=21636 \times 6 = 21636×6=216, and finally multiply 216×6=1296216 \times 6 = 1296216×6=1296.
- So, 64=12966^4 = 129664=1296, and multiplying two such results gives 1296×1296=1,679,6161296 \times 1296 = 1,679,6161296×1296=1,679,616.
- For 686^868, the same calculation yields 68=1,679,6166^8 = 1,679,61668=1,679,616.
Thus, multiplying 64×646^4 \times 6^464×64 gives you 686^868, which is a simpler, more elegant way to express the result.
