Advice 1: How to solve examples with roots

The root of degree n of the number called is a number that when raised to a this degree will give the number from which to extract the root. Most often, actions are performed with the square roots, which correspond to 2 degrees. At root it is often impossible to find it explicitly, and the result is a number that cannot be expressed in the form of a genuine fraction (transcendental). But using some techniques, we can greatly simplify the solution of examples with roots.
How to solve examples with roots
You will need
  • - the notion of root of a number;
  • - actions with degrees;
  • formulas of reduced multiplication;
  • calculator.
If you do not need absolute accuracy, while solving examples with the roots use a calculator. To extract the number of the square root, enter it on the keyboard, and just press the relative button, which shows the sign of the root. As a rule, calculators, take the square root. But for calculation of roots of higher degrees, use of a number raised to the degree (engineering calculator).
To extract the square root raise to the power 1/2, cubic root of 1/3, and so on. In this case you must keep in mind that when extracting the roots of even degree, the number must be positive, otherwise the calculator will not just give the answer. This is due to the fact that during the construction in even degree any number will be positive, for example, (-2)^4=(-2)∙ (-2)∙ (-2)∙ (-2)=16. To extract the square root evenly, when possible, use the table of squares of natural numbers.
If not near a calculator, or require absolute precision in the calculations use properties of roots and various formulas to simplify expressions. Many of the numbers, you can remove the root part. To do this, use the property that the root of the product of two numbers is equal to the product of the roots of these numbers √m * n=√m∙√n.
Example. Calculate the value of the expression (√80-√45)/ √5. Direct calculation will give nothing, because evenly is not removed no root. Convert the expression (√16∙5-√9∙5)/ √5=(√16∙√5-√9∙√5)/ √5=√5∙(√16-√9)/ √5. Perform a reduction of the numerator and denominator by √5, get (√16-√9)=4-3=1.
If the radical expression or the root raised to the power, when you root, use the property that the exponent radical expressions can be divided into the degree of the root. If the division is performed evenly, the number is entered from under the root. For example, √5^4=52=25.

Example. To calculate the value of the expression (√3+√5)∙(√3-√5). Apply the formula of difference of squares and get the (√3)2-(√5)2=3-5=-2.

Advice 2 : How to make a number of root

In most cases, it's easier to calculate on the calculator radical expression. But if you want to solve the problem in General or radical expression contains the unknown variables or the conditions of the problem it is necessary only to simplify, not to calculate, will have to find ways of making any number of under root.
How to make a number of root
Use the definition of rootas a mathematical operation, from which it follows that the extraction of a root is the inverse operation of raising a number to a power. This means that the number can be taken from under the root if the decrease in radical expressions , the number of times that corresponds to a raised to the power passed. For example, to take from under the square root of the number 10, divide remaining under the root expression of ten squared.
Pick up the radical number of this multiplier, making of which is under root do simplify the expression - otherwise the operation will lose meaning. For example, if under the sign of the root with exponent equal to three (cubic root), is the number 128, then out of sign can be taken, for example, the number 5. In this radical , the number 128 will have to be divided by 5 cubed: 3√128 = 5∗3√(128/53) = 5∗3√(128/125) = 5∗3√1.024. If the presence of fractional numbers under the sign of root is not contrary to the conditions of the problem, the solution can be left in this form. If you need a more simple variant, we first divide radical expression for such an integer multiplier, cube root one of which will be a whole numberC. for Example: 3√128 = 3√(64∗2) = 3√(43∗2) = 4∗3√2.
Use for the selection of the multipliers radical number calculator, if you calculate in the mind the powers of a number is not possible. This is especially important for the rootm with the exponent greater than two. If you have Internet access, you can calculate the built-in Google search engine and Nigma solvers. For example, if you need to find the largest integer multiplier, which can be taken from under the sign of the cubic root for numbers 250, then going to Google enter "6^3" to check, can you just take out of sign of root six. The search engine displays the result equal to 216. Alas, the 250 cannot be divided without a remainder is the number. Then enter the query 5^3. The result will be 125, and it allows you to break 250 on the multipliers 125 and 2, and therefore to stand under the sign of the root number 5, leaving the number 2.

Advice 3 : How to fold the root and the number

Arithmetic root of n-th degree of a real number a is called non-negative number x, n-I the degree of which is equal to the number a. Ie (√n) a = x, x^n = a. There are various ways of addition and the arithmetical root of a rational number. Here for greater clarity, will be considered the roots of the second degree (or square roots), explanations will be complemented by examples of calculation of roots of other degrees.
How to fold the root and the number
Let the set of expressions of the form a + √b. The first thing to do is to determine whether the number b is a perfect square. Ie try to find such a number c that c^2 = b. In this case, you extract the square root of the number b, get c and put him with number a: a + √b = a + √(c^2) = a + S. If you are not dealing with the square root, and the root of n-th degree, then complete extraction of the number b from under the sign of the root is necessary to this number was the nth degree of a certain number. For example, the number 81 will be removed from under the square root: √81 = 9. Also it will be removed from under the sign of the fourth root: (√4) 81 = 3.
Note the following examples.

• 7 + √25 = 7 + √(5^2) = 7 + 5 = 12. Here under the sign of the square root is the number 25, which is the full square of 5.

• 7 + (√3)27 = 7 + (√3) (3^3) = 7 + 3 = 10. Here were extracted the cube root of the number 27 which is the cube of the number 3.

• 7 + √(4/9) = 7 + √( (2/3)^2 ) = 7 + 2/3 = 23/3. To extract the square root of a fraction you must remove the root from the numerator and the denominator.
If the b number under the root sign is not a perfect square, then try to decompose it into factors and make the multiplier, which is a complete square, under the sign of the root. Ie let the number of b has the form b = c^2 * d. Then √b = √(c^2 * d) = c * √d. Or b can contain the squares of two numbers, i.e. b = c^2 * d^2 * e * f . Then √b = √(c^2 * d^2 * e * f) = c * d * √(e * f).
Examples of sentencing multipliers under the sign of the root:

• 3 + √18 = 3 + √(3^2 * 2) = 3 + 3√2 = 3 * (1 + √2).

• 3 + √(7 / 4) = 3 + √ (7 / 2^2) = 3 + √7 / 2 = (6 + √7) / 2. In this example, was handed down the full square of the denominator of a fraction.

• 3 + (√4)240 = 3 + (√4) (2^4 * 3 * 5) = 3 + 2 *(√4) 15. There turned out to take 2 in the fourth degree under the sign of the fourth root.
And finally, if you need to obtain the approximate result (if the radical expression is a perfect square), use the calculator to compute the value of the root. For example, 6 + √7 ≈ 6 + 2,6458 = 8,6458.
Useful advice
To calculate the approximate value of the root on standard calculator remember to take the square root of a number is equivalent to raising a number to the power 1/2. Similarly, the extraction of the cube root is equivalent to raising a number to the power 1/3, the fourth root – the construction of the power 1/4, etc.
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