I need to use rational exponents to write the answer to:
the cube root of x times the 4 root of 7x.
there is a rule when the numbers on the radicand are different, yes?
I multiply the sq rt numbers to get 12 yes? and then do I multiply the x times 7x to get 7x^7?
x^(1/3) * (7x)^(1/4)
= x^(1/3) * 7^(1/4) * x^(1/4)
= 7^(1/4) * x(1/3 + 1/4)
= 7^(1/4) * x (7/12)
In other words, when you multiply you add exponents
think
10^1 * 10^2 = 10^3 = 1,000
like you know that
10 * 100 = 1000
Yes, there is a rule for simplifying expressions with different numerical radicands using rational exponents.
To write the expression in terms of rational exponents, you can start by using the fact that any nth root can be written as a fractional exponent.
1. Write the cube root of x as x^(1/3).
2. Write the 4th root of 7x as (7x)^(1/4).
Now, to find the product of these two terms, you can use the rule that when multiplying variables with the same bases, you add the exponents:
x^(1/3) * (7x)^(1/4)
We can also simplify the expression further by multiplying the exponents:
x^(1/3) * (7x)^(1/4) = x^((1/3) + (1/4))
To add the exponents, you'll need to find a common denominator. The least common multiple (LCM) of 3 and 4 is 12.
1/3 = 4/12
1/4 = 3/12
Now we can rewrite the expression:
x^((4/12) + (3/12)) = x^(7/12)
Therefore, the expression "the cube root of x times the 4th root of 7x" can be written as x^(7/12) using rational exponents.