Please help..... give me step by step answer so I understand better
Simplify by rationalizing the denominator ....15 sqrt 2/ sqrt 270
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To simplify the expression by rationalizing the denominator, we need to eliminate the square root from the denominator by multiplying both the numerator and denominator by an appropriate value.
Here are the step-by-step instructions:
Step 1: Simplify the numerator if possible. In this case, the numerator is already simplified as 15√2.
Step 2: Simplify the denominator if possible. Here, we have the square root of 270. To simplify it, we can find the perfect squares that are factors of 270. We see that 270 can be written as 9 * 30, and 9 is a perfect square (3 * 3). So, the square root of 270 can be simplified to 3√30.
Step 3: Now, we have the expression as (15√2) / (3√30).
Step 4: To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate is found by changing the sign of the middle term in the denominator.
In this case, the conjugate of 3√30 is -3√30.
Step 5: Multiply the numerator and denominator by the conjugate. We get:
((15√2) / (3√30)) * ((-3√30) / (-3√30))
Simplifying this expression, we get:
(-45√2√30) / (-9 * 30)
Here, √2 * √30 = √(2 * 30) = √60 = 2√15.
So, the expression becomes:
(-45 * 2√15) / (-270)
Simplifying further:
(-90√15) / (-270)
Step 6: Simplify the negative signs. Since we have a negative divided by a negative, they cancel each other out, so the expression becomes:
90√15 / 270
Step 7: Divide the numerator and denominator by their greatest common factor, which is 90 in this case. We get:
√15 / 3
Therefore, the simplified expression after rationalizing the denominator is √15 / 3.
15√2/√270 * √270/√270 15√540/270 = 15*6√15/270 = √15/3
or
15√2/√270 = 15/√135 = 15/3√15 = 5/√15 * √15/√15 = 5√15/15 = √15/3