how to simplify 784^2/3

[stanley92]

Hi, I am getting confused with part of the simplification of 784^2/3

I understand the first step: (2^4 x 7^2)^2/3 however i do not understand how you get to the next stage which is:

[FONT=&quot][/FONT]

[FONT=MathJax_Size2]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Size2])[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT]​

[FONT=&quot]and the simplified surd is therefore[/FONT]

[FONT=&quot][/FONT]

[FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]​

Please can someone help me understand the last 2 steps. I have tried looking this up but cannot find an explanation of how to multiple an exponent with another exponent which is a fraction.

Thanks

Thanks


[FONT=&quot]

​

​

[FONT=MathJax_Main]
[/FONT]
[/FONT][FONT=MathJax_Main][/FONT]

 

[tkhunny]

Hi, I am getting confused with part of the simplification of 784^2/3

I understand the first step: (2^4 x 7^2)^2/3 however i do not understand how you get to the next stage which is:

[FONT=MathJax_Size2]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Size2])[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT]​

and the simplified surd is therefore

[FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]​

Please can someone help me understand the last 2 steps. I have tried looking this up but cannot find an explanation of how to multiple an exponent with another exponent which is a fraction.

Thanks

I never have understood the concept of counting steps.

Are there any perfect cubes in \(\displaystyle 2^{4}\cdot 7^{2}\)

 

[topsquark]

Please can someone help me understand the last 2 steps. I have tried looking this up but cannot find an explanation of how to multiple an exponent with another exponent which is a fraction.

Thanks

Fractional exponents follow the same rules as integers. \(\displaystyle \left ( x^a \right ) ^b = x^{ab}\). (This also works for exponents that are real or complex.)

So \(\displaystyle \left ( 2 ^4 \right ) ^{ 2/3 } = 2^{ ( 4 \cdot 2/3 ) } = 2^{ 8/3 }\).

-Dan

 

[JeffM]

Fractional exponents follow the same rules as integers. \(\displaystyle \left ( x^a \right ) ^b = x^{ab}\). (This also works for exponents that are real or complex.)

So \(\displaystyle \left ( 2 ^4 \right ) ^{ (2/3) } = 2^{ ( 4 \cdot 2/3 ) } = 2^{ 8/3 }\).

-Dan

Dan, I think the OP understands fractional exponents.

\(\displaystyle 784^{2/3} = WHAT?\)

is the problem (I think) despite the missing parentheses on the exponent.

\(\displaystyle 784 = 2 * 392 = 2 * 2 * 196 = 2^2 * 2 * 98 =\)

\(\displaystyle 2^3 * 2 * 49 = 2^4 * 7^2.\)

The OP's 24 X 72 means (I think) 2^4 * 7^2.

I can follow what is going on to here despite the lack of proper exponents. Student was factoring by primes. Good show.

Here is where I think he is getting mixed up.

\(\displaystyle (2^4 * 7^2)^2 = 2^8 * 7^4 = 2^6 * 2^2 * 7^3 * 7 = (2^2 * 7)^3 * 2^2 * 7 =\)

\(\displaystyle (4 * 7)^3 * 4 * 7 = 28^3 * 28.\)

\(\displaystyle \therefore 784^{2/3} = \sqrt[3]{784^2} = \sqrt[3]{28^3 * 28} \implies\)

\(\displaystyle \therefore 784^{2/3} = \sqrt[3]{28^3} * \sqrt[3]{28} = 28\sqrt[3]{28} = 28 * 28^{1/3}.\)

EDIT: Stanley, I hope this clears up your confusion, but you must put your fractional exponents in parentheses when typing. And you ABSOLUTELY MUST indicate exponentiation by using the ^ symbol (shift on the 6 on a normal keyboard.)

What you meant to write in the last line was 28 * 28^(1/3).

 

[Dr.Peterson]

Hi, I am getting confused with part of the simplification of 784^2/3

I understand the first step: (2^4 x 7^2)^2/3 however i do not understand how you get to the next stage which is:

[FONT=MathJax_Size2]([/FONT][FONT=MathJax_Main]2[/FONT]^{[FONT=MathJax_Main]4[/FONT]}[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT]^{[FONT=MathJax_Main]2[/FONT]}[FONT=MathJax_Size2])[/FONT]^{[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]}[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT]^{[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]}[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT]^{[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]}[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT]^{[FONT=MathJax_Main]2[/FONT]}[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT]^{[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]}[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]7[/FONT]^{[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]}[FONT=MathJax_Main],[/FONT]
​

and the simplified surd is therefore

[FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]28[/FONT]^{[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]3[/FONT]}
​

Please can someone help me understand the last 2 steps. I have tried looking this up but cannot find an explanation of how to multiple an exponent with another exponent which is a fraction.

First, a comment on notation: fractional exponents, when written with inline "^" notation, require parentheses: 784^(2/3).

Second, I formatted the exponents in the work you show to make it readable.

What they have done is first to distribute the exponent, but they didn't write the result of that first step: (2⁴)^2/3×(7²)^2/3.

Then they applied the rule that a power raised to a power multiplies the exponents, resulting in 2^4×2/3×7^2×2/3. (They wrote it in a way that didn't quite make that clear.)

Then they skipped another step: the multiplications give 4×2/3 = 8/3 = 2 2/3, and 2×2/3 = 4/3 = 1 1/3 as mixed numbers. That leaves you with 2^2+2/3×7^1+1/3.

Then, applying the rule that adding exponents is equivalent to multiplying the results, they get the next thing they wrote, 2²×2^2/3×7¹×7^1/3.

Now, there are lots of ways to write a final result; what they did is to combine the two integer powers as 2²×7¹ = 4×7 = 28, and the fractional powers as 2^2/3×7^1/3 = (2²)^1/3×7^1/3 = (2²×7)^1/3 = (28)^1/3.

Clearly, they are expecting you to follow this without having to be shown each bit. (The "steps" are just the steps they took, and they didn't show half of them. You can take any steps you want, as long as they are all valid.) You may want to go back and read through some simpler examples where they would have shown, and maybe explained, everything they did, until you know the properties of exponents well enough to recognize them when they are not shown in great detail. Once you are expert, you can write less, as they did. I don't recommend it, though.