ACT Math : Square Roots and Operations
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Example Questions
Example Question #1 : Square Roots And Operations
Find the product:
Simplify the radicals, then multiply:
Example Question #2 : Square Roots And Operations
Simplify the following completely:
To simplify this expression, simply multiply the radicands and reduce to simplest form.
Example Question #1 : How To Multiply Square Roots
Simplify:
When multiplying square roots, the easiest thing to do is first to factor each root. Thus:
Now, when you combine the multiplied roots, it will be easier to come to your final solution. Just multiply together everything "under" the roots:
Finally this can be simplified as:
Example Question #4 : Square Roots And Operations
Simplify the following:
When multiplying square roots, the easiest thing to do is first to factor each root. Thus:
Now, when you combine the multiplied roots, it will be easier to come to your final solution. Remember that multiplying roots is very easy! Just multiply together everything "under" the roots:
Finally this can be simplified as:
Example Question #5 : Square Roots And Operations
State the product:
Don't try to do too much at first for this problem. Multiply your radicals and your coefficients, then worry about any additional simplification.
Now simplify the radical.
Example Question #6 : Square Roots And Operations
Find the product:
Don't try to do too much at first for this problem. Multiply your radicals and your coefficients, then worry about any additional simplification.
Now, simplify your radical.
Example Question #1 : How To Find A Ratio Of Square Roots
x4 = 100
If x is placed on a number line, what two integers is it between?
3 and 4
2 and 3
5 and 6
Cannot be determined
4 and 5
3 and 4
It might be a little difficult taking a fourth root of 100 to isolate x by itself; it might be easier to select an integer and take that number to the fourth power. For example 34 = 81 and 44 = 256. Since 34 is less than 100 and 44 is greater than 100, x would lie between 3 and 4.
Example Question #3 : Square Roots And Operations
What is the ratio of 
The ratio of two numbers is merely the division of the two values. Therefore, for the information given, we know that the ratio of
can be rewritten:
Now, we know that the square root in the denominator can be "distributed" to the numerator and denominator of that fraction:
Thus, we have:
To divide fractions, you multiply by the reciprocal:
Now, since there is one 
This gives you:
Rationalize the denominator by multiplying both numerator and denominator by 
Let's be careful how we write the numerator so as to make explicit the shared factors:
Now, reduce:
This is the same as
Example Question #1 : Square Roots And Operations
What is the ratio of 
To find a ratio like this, you need to divide 
Next, you can write the ratio of the two variables as:
Now, when you divide by a fraction, you can rewrite it as the multiplication by the reciprocal. This gives you:
Simplifying, you get:
You should rationalize the denominator:
This is the same as:
Example Question #4 : Square Roots And Operations
Find the sum:
Find the Sum:
Simplify the radicals:
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