How do I simplify 2^(-m) x 3^(-m) x 6^(2m) x 3^(2m) x 2^(2m) as a power of 6?

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In summary, the conversation is about a person asking for help with a math problem for their niece. They have tried to solve it but are stuck at a point where they need to get rid of base 2 and base 3 terms. The solution is 6^(3m) but they are unsure how to get there. They are reminded that 2 times 3 is equal to 6.
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Darkmisc
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Homework Statement
Write 2^(-m) x 3^(-m) x 6^(2m) x 3^(2m) x 2^(2m) as a power of 6.
Relevant Equations
Write 2^(-m) x 3^(-m) x 6^(2m) x 3^(2m) x 2^(2m) as a power of 6.
Hi everyone

Could someone please help me with a yr 10 maths problem? It's for my niece. I've done 2nd yr uni maths and can't seem to solve it.

Write 2^(-m) x 3^(-m) x 6^(2m) x 3^(2m) x 2^(2m) as a power of 6.

I've attached my attempt in the file. I get stuck at the point where I need to get rid of the base 2 and base 3 terms.

The answer is 6^(3m), but I don't know how to get there. Thanks
 

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  • #2
Darkmisc said:
I've done 2nd yr uni maths
Then you should know hat ##2\times 3=6## !
 
  • #3
Thanks. It's been a while since I've done maths.
 
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1. What is the first step in simplifying this expression?

The first step is to group the like terms together. In this case, we can group the terms with a base of 2 and the terms with a base of 3.

2. How do I simplify the terms with a base of 2?

To simplify the terms with a base of 2, we can use the rule that states when multiplying powers with the same base, we add the exponents. So, 2^(-m) x 6^(2m) x 2^(2m) becomes 2^(-m + 2m + 2m) which simplifies to 2^(3m).

3. What about the terms with a base of 3?

Similarly, for the terms with a base of 3, we can use the same rule and add the exponents. So, 3^(-m) x 3^(2m) becomes 3^(-m + 2m) which simplifies to 3^(m).

4. How do I simplify the remaining terms?

Now that we have simplified the terms with a base of 2 and 3, we are left with 2^(3m) x 3^(m) x 6^(m). To simplify this further, we can use the rule that states when raising a power to another power, we multiply the exponents. So, 6^(m) can be written as (2 x 3)^(m) which becomes 2^(m) x 3^(m). Therefore, our expression becomes 2^(3m) x 3^(m) x 2^(m) x 3^(m).

5. What is the final simplified expression?

Finally, we can use the rule that states when multiplying powers with the same base, we add the exponents. So, our final simplified expression becomes 2^(3m + m) x 3^(m + m) which simplifies to 2^(4m) x 3^(2m). Therefore, the given expression, 2^(-m) x 3^(-m) x 6^(2m) x 3^(2m) x 2^(2m), can be written as a power of 6 as 6^(2m).

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