a. √ 24 + √24
b 10√42
c. 10√21
d. √ 52
The simplified expressions for the given options are:
a. √24 + √24 = 2√24 = 2√(4 * 6) = 2√4√6 = 2 * 2√6 = 4√6
b. 10√42 cannot be simplified further.
c. 10√21 cannot be simplified further.
d. √52 = √(4 * 13) = √4√13 = 2√13
To simplify each expression step-by-step:
a. √24 + √24
First, we simplify each square root separately:
√24 = √(4 × 6) = √4 × √6 = 2√6
Therefore:
√24 + √24 = 2√6 + 2√6 = 4√6
b. 10√42
We can't simplify this radical any further since 42 does not have any perfect square factors.
c. 10√21
Similar to the previous example, this radical cannot be simplified further.
d. √52
First, we find the largest perfect square that divides evenly into 52, which is 4.
√52 = √(4 × 13) = √4 × √13 = 2√13
So, the simplified expressions are:
a. 4√6
b. 10√42
c. 10√21
d. 2√13
To simplify the given expressions, we can start by finding the prime factors of the number under each square root symbol. Let's break down each expression one by one:
a. √24 + √24:
To simplify √24, we need to break it down into its prime factors: 2 * 2 * 2 * 3. Since we have two identical square roots, we can combine them:
√24 + √24 = 2√6 + 2√6
Next, we can simplify the expression by combining like terms:
2√6 + 2√6 = 4√6
So the simplified answer is 4√6.
b. 10√42:
To simplify √42, we need to find its prime factors: 2 * 3 * 7. We can then combine the square root and the coefficient:
10√42 = 10 * √(2 * 3 * 7) = 10 * √(2 * 3) * √7 = 10 * √6 * √7
Lastly, we can multiply the coefficients and combine the square roots:
10 * √6 * √7 = 10√(6 * 7) = 10√42
So the given expression is already simplified.
c. 10√21:
To simplify √21, we need to find its prime factors: 3 * 7. We can then combine the square root and the coefficient:
10√21 = 10 * √(3 * 7) = 10 * √3 * √7
Lastly, we can multiply the coefficients and combine the square roots:
10 * √3 * √7 = 10√(3 * 7) = 10√21
So the given expression is already simplified.
d. √52:
To simplify √52, we need to find its prime factors: 2 * 2 * 13. We can then separate the square root:
√52 = √(2 * 2 * 13) = √(2 * 2) * √13 = 2√13
So the simplified answer is 2√13.
In summary:
a. √24 + √24 = 4√6
b. 10√42 remains as it is.
c. 10√21 remains as it is.
d. √52 = 2√13