Simplifying Radicals Finding Hidden Perfect Squares And Taking Their Root? 97 Most Correct Answers

is one that contains a root (square root, cube root, etc.). In Algebra 1, radical expressions are mainly restricted to square root (a = 2) and cube root (a = 3) expressions. Let’s examine square roots: Let’s examine square roots:

Square roots have an index value of two. If you see a radical with no index listed, it is assumed to be an index of two, a square root.

A square root is, in its simplest form, if

1. The radicand does not contain perfect square factors

2. the radicand is not a fraction

3. there are no radicals in the denominator of a fraction.

1. Find the greatest perfect square factor (the greatest perfect square that is evenly divisible by 48). You must be familiar with a list of perfect squares.

2. Give each factor its own root sign. 3. Reduce the resulting “perfect square” radical. 4. ANSWER:

Don’t worry if you don’t choose the BIGGEST square factor to start with. You can still get the correct answer, but you have to repeat the process. See what happens when we choose 4 instead of 16 to start:

Notice how the Out-Front is multiplied by 2 in the second row for the rest of the problem.

if you don’t choose the LARGEST square factor to start with. You can still get the correct answer, but you have to repeat the process. See what happens if we choose 4 instead of 16 to begin with: Notice how the first 2 in the second row is multiplied for the rest of the problem.

The number 23 cannot be factored by any of the perfect squares (23 is a prime number). This is a trick question as it is already in its simplest form and cannot be reduced any further.

1. Enter your own root symbols for the numerator and denominator.

2. Multiply the numerator and denominator by a radical, eliminating the radical in the denominator by forming a perfect square under the radical. If no smaller value can be found, multiply by the same root value that is in the denominator, which automatically creates a perfect square.

Give the numerator and denominator their own radical signs. Multiply that by a radical that eliminates the radical in the denominator by making a perfect square under the radical. If no smaller value can be found, multiply by the same root value that is in the denominator, which automatically creates a perfect square. This process of removing a radical from the denominator is called “rationalizing the denominator” because it converts the denominator to a rational (not irrational) value.

The square root of a value is a quantity squared equal to the radicand (the number under the square root symbol). For example, the square root of 16 could be either +4 or -4, since both squared equals 16.

However, it should be understood that the square root (radical) symbol designates only the positive root, which is called the “principal square root”. The value of a value is a quantity squared equal to the radicand (the number under the square root symbol). For example, the square root of 16 could be either +4 or -4 since both squared equal 16. However, it should be understood that die designates only the root referred to as die. When you solve the equation x2 = 25, look for both solutions: +5 and -5. So we write:

Let’s see how this approach works with Cube Roots:

1. Find the largest perfect cube factor (the largest perfect cube that is evenly divisible by 24).

2. Give each factor its own root sign. 3. Reduce the resulting “perfect cube” radical. 4. ANSWER:

Perfect Squares

4 = 2×2

9 = 3×3

16 = 4×4

25 = 5×5

36 = 6×6

49 = 7×7

64 = 8×8

81 = 9×9

100 = 10×10

121 = 11×11

144 = 12×12

169 = 13×13

196 = 14×14

225 = 15×15

square roots

prime factorization

Radicals can also be simplified by expressing the radicand by prime factorization and looking for groups of two similar factors to form a perfect square factor.

product rule

where a ≥ 0, b ≥ 0

“The square root of a product is equal to the product of the square roots of each factor.”

This theorem allows us to apply our radical simplification method.

quotient rule

where a ≥ 0, b > 0 “The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator.”

Perfect dice

8 = 2x2x2

27 = 3x3x3

64 = 4x4x4

125 = 5x5x5

cube roots

Remember:

When working with radicals, the term “simplify” means to find an equivalent expression.

It doesn’t mean finding a decimal approximation.

radical

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Use the product property to simplify radical expressions

We will simplify root expressions in a similar way that we simplified fractions. A fraction is simplified if the numerator and denominator have no common divisors. To simplify a fraction, we look for common factors in the numerator and denominator.

A radical expression is considered simplified if it has no factors of . So, to simplify a radical expression, we look for factors in the radicand that are powers of the index.

Simplified radical expression For real numbers a and m and

For example, considered simplified because there are no perfect square factors in 5. But not simplified because 12 has a perfect square factor of 4.

Similarly, it simplifies because there are no perfect cube factors in 4. But not simplified because 24 has a perfect cubic factor of 8.

To simplify root expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know the equivalent of Product Property of Roots says so

Product property of nth roots If and are real numbers and then is an integer

We use the product property of roots to remove all perfect square factors from a square root.

Simplify square roots using Roots Simplify product property:

Simplify:

Simplify:

In the previous example, notice that the simplified form of is is the product of an integer and a square root. We always write the whole number in front of the square root.

Be careful to write your integer in a way that doesn’t get confused with the index. The expression is very different from

Simplify a radical expression with the product property. Find the greatest divisor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors using this factor. Use the product rule to rewrite the radical as the product of two radicals. Simplify the root of perfect power.

We will use this method in the next example. It can be helpful to have a table of perfect squares, cubes, and fourth powers.

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

The next example is very similar to the previous examples, but with variables. Don’t forget to use the absolute value sign when taking a straight root in a one-variable expression in the radical.

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

We do the same thing when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

In the next example, we continue to use the same methods even though there is more than one variable under the radical.

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ

Simplify: ⓐ ⓑ ⓐ ⓑ

Simplify: ⓐ ⓑ ⓐ ⓑ

Simplify: ⓐ ⓑ ⓐ ⓑ not a real number

We have seen how to use the order of operations to simplify some expressions involving radicals. In the next example, we have the sum of an integer and a square root. We simplify the square root, but we cannot add the resulting expression to the integer because one term contains a radical and the other does not. The next example also contains a root fraction in the numerator. Remember that to reduce a fraction, you need a common factor in the numerator and denominator.

Simplify: ⓐ ⓑ ⓐ The terms cannot be added because one has a radical and the other does not. Trying to add an integer and a radical is like trying to add an integer and a variable. They are not like terms! ⓑ

Simplify: ⓐ ⓑ ⓐ ⓑ

The square root of a number is a number that, when multiplied by itself, gives the desired value. For example, the square root of 49 is 7 (7×7=49). Multiplying a number by itself is called squaring.

Numbers whose square roots are integers (or, more accurately, positive integers) are called perfect squares. Numbers with decimals are not perfect square roots.

All positive numbers have a positive square root number, called the principal number, and a negative number. These numbers are all known as real numbers.

All negative numbers have a complex number as the square root. A complex number is a number multiplied by i. i is the “imaginary” square root of -1. It’s called imaginary, but it exists for mathematicians.

How do you write square roots?

A square root equation is written with a root sign or symbol (). The number we want to take the square root of comes after or below the rest of the radical (e.g. 3 if we wanted to find the square root of 3). The number after the radical is called the radican. On a calculator, you might see “sqrt” instead of the radical.

What do we use square roots for?

It might be a little hard to visualize, but square roots are some of the most useful numbers out there. Square root functions are super important for physics equations of all kinds. They’re also valuable for statistics; Statisticians use square roots all the time to analyze the correlation between different data points.

List of perfect squares

Use this table to find the squares and square roots of numbers from 1 to 100.

You can also use this table to estimate the square roots of larger numbers.

For example, if you want to find the square root of 2000, look in the middle column until you find the number closest to 2000. The number in the middle column closest to 2000 is 2,025 .

, look up the column until you find the number closest to 2000. The number in the middle column closest to 2000 is . Now look at the number to the left of 2,025 to find its square root. The square root of 2,025 is 45.

to find its square root. The square root of 2,025 is . Therefore, the approximate square root of 2,000 is 45.

To get a more accurate number, you’ll need to use a calculator (44.721 is the more accurate square root of 2,000).

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Number Square Square Root 1 1 1 1 1,414 3 9 1.732 4 16 2.236 6 36 2.446 8 64 2.828 9 81 3.162 11 121 3.464 13 169 3.606 14 196 3.742 3.873 16 256 4,000 17 289 4.123 18 361 4.359 20 400 4.583 22 484 4.796 24 576 4.899 25 676 5.096 28 784 5.241 5.477 5.568 32 1.024 5. 33 1,089 5.745 34 1,156 5.831 35 1,225 5.916 36 1,296 6.000 37 1,369 6.083 38 1,444 6.164 39 1,521 6.245 40 1,600 6.325 41 1,681 6.403 42 1,764 6.481 43 1,849 6.557 44 1,936 6.633 45 2,025 6.708 46 2,116 6.782 47 2,209 6.856 48 2,304 6.928 49 2,401 7.000 50 2,500 7.071 51 2,601 7.141 52 2,704 7.211 53 2,809 7.280 54 2,916 7.348 55 3,025 7.416 56 3,136 7.483 57 3,249 7.550 58 3,364 7.616 59 3,481 7.681 60 3,600 7.746 61 3,721 7.810 62 3,844 7.874 63 3,969 7.937 64 4,096 8.000 65 4,225 8.062 66 4,356 8,124 67 4,489 8,185 68 4,624 8,246 69 4,761 8,307 70 4,900 8,367 71 5.04 1 8.426 72 5,184 8.485 73 5,329 8.544 74 5,476 8.602 75 5,625 8.660 76 5,776 8.718 77 5,929 8.775 78 6,084 8.832 79 6,241 8.888 80 6,400 8.944 81 6,561 9.000 82 6,724 9.055 83 6,889 9.110 84 7,056 9.165 85 7,225 9.220 86 7,396 9.274 87 7,569 9.327 88 7,744 9.381 89 7,921 9.434 90 8,100 9.487 91 8,281 9.539 92 8,464 9.592 93 8,649 9.644 94 8,836 9.695 95 9,025 9.747 96 9,216 9.798 97 9,409 9.849 98 9,604 9.899 99 9,801 9.950 100 10,000 10.000 NOTE: Square roots in this table are rounded to the next thousandth.