It says here Write as a single power, then evaluate. Express answers in rational form.
⁴√18× ⁴√9 ÷ ⁴√2
18^(1÷4)× 9^(1÷4) ÷2^(1÷4)
My problem here is I couldn't find a way to get all the numbers to have the same base so I could continue to the answer please help
since all 3 factors have an exponent of 1/4
you can say:
⁴√18× ⁴√9 ÷ ⁴√2
= (18x9÷2)^(1/4)
= 81^(1/4)
= 3
(one of the basic exponent rules:
(a^n)(b^n) = (ab)^n
To express all the numbers with the same base, let's rewrite them using exponentials:
18^(1/4) × 9^(1/4) ÷ 2^(1/4)
Now, we can simplify each term by raising them to the 4th power:
(18^(1/4))^4 × (9^(1/4))^4 ÷ (2^(1/4))^4
This simplifies to:
18^(1/4 × 4) × 9^(1/4 × 4) ÷ 2^(1/4 × 4)
Which becomes:
18^1 × 9^1 ÷ 2^1
Simplifying further:
18 × 9 ÷ 2
Finally, evaluating:
(18 × 9) ÷ 2 = 162 ÷ 2 = 81
To simplify the expression ⁴√18 × ⁴√9 ÷ ⁴√2, we need to find a common base for all the numbers within the fourth roots.
Let's rewrite each number with the same base:
18 = 2 × 9
Now, we can rewrite 18^(1÷4) as (2 × 9)^(1÷4).
Recall the rule of exponents: (ab)^c = a^c × b^c.
Applying this rule to our expression, we have:
(2 × 9)^(1÷4) = 2^(1÷4) × 9^(1÷4)
Now our expression is:
2^(1÷4) × 9^(1÷4) × ⁴√9 ÷ ⁴√2
With this step, we have a common base of 9 within the fourth root terms.
So ⁴√9 can be written as (9^(1÷4))^4.
Now our expression is:
2^(1÷4) × 9^(1÷4) × (9^(1÷4))^4 ÷ ⁴√2
Since (9^(1÷4))^4 simplifies to 9, we can rewrite the expression as follows:
2^(1÷4) × 9^(1÷4) × 9 ÷ ⁴√2
Now we have a common base of 2 within the fourth root term.
So ⁴√2 can be written as (2^(1÷4))^2.
The expression now becomes:
2^(1÷4) × 9^(1÷4) × 9 ÷ (2^(1÷4))^2
Since (2^(1÷4))^2 simplifies to 2^(1÷2) or √2, we can rewrite the expression as follows:
2^(1÷4) × 9^(1÷4) × 9 ÷ √2
By multiplying the numbers outside the roots, we get:
2^(1÷4) × 9^(1÷4) × 9 = 2^(1÷4) × 3^(2÷4) × 9
Simplifying the exponents:
2^(1÷4) × 3^(2÷4) × 9 = 2^(1/4) × 3^(1/2) × 9
To evaluate the expression further, we can calculate the values of the exponents:
2^(1/4) = √2
3^(1/2) = √3
Substituting these values back into our expression:
√2 × √3 × 9 = √(2 × 3) × 9 = 3√6 × 9 = 27√6
Therefore, the simplified single power form of the given expression is 27√6.