Properties of Roots of Real Numbers

In summary, the properties of roots of real numbers include the product property, quotient property, power property, radical property, and distributive property. To simplify a radical expression, you can factor and identify perfect squares before simplifying. A square root is the inverse of squaring a number, while a cube root is the inverse of cubing a number. To solve equations involving roots, you can isolate, square, and solve for the variable. And finally, while you can take the square root or cube root of a negative number, the result will be a complex number.
  • #1
IndigoSwing4
3
0
I have three problems that I can't seem to solve. I was wondering if anybody could help me or explain to me how to solve these. Note: * = multiplication.

1. (6^1/2 * 2^1/3)^6

2. ^4√7 + 2^4√1792

3. 3(X-4)^1/2 + 5 = 11
 
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  • #2
#2 seems to be missing a term. What is raised to 4 times square root of 7?

#3 seems easy, I think. It reduces to: [tex]\sqrt{x-4}=2.[/tex]
That gives x= 8, which works.

For #1, use the binominal expansion: [tex](a+b)^6= a^6+6a^5b+15a^4b^2 +20a^3b^3+15a^2b^4+6ab^5+b^6.[/tex]
 
Last edited:
  • #3
1. (6^(1/2)*2^(1/3))^6

That's "*", not "+", so you don't need the formula robert Ihnot gave:

(6^(1/2)*2^(1/3))^6= (6^(1/2))^6*(2^(1/3))^6
= 6^((1/2)*6)*2^((1/3)*6)= 6^3*2^2= 216*4= 864.
 

1. What are the properties of roots of real numbers?

The properties of roots of real numbers include the following:

  • Product Property: The nth root of a product is equal to the product of the nth roots of the factors. In other words, ∛(ab) = ∛a * ∛b.
  • Quotient Property: The nth root of a quotient is equal to the quotient of the nth root of the numerator and the nth root of the denominator. In other words, ∛(a/b) = (∛a) / (∛b).
  • Power Property: The nth power of a number is equal to the nth root of that number raised to the nth power. In other words, (∛a)^n = a.
  • Radical Property: The nth root of a number raised to the nth power is equal to the original number. In other words, ∛(a^n) = a.
  • Distributive Property: The nth root of a sum or difference is not equal to the sum or difference of the nth roots of each term. In other words, ∛(a + b) ≠ ∛a + ∛b.

2. How do you simplify radical expressions?

To simplify a radical expression, you can follow these steps:

  1. Factor the number under the radical sign into its prime factors.
  2. Identify any perfect squares (numbers that have a square root that is a whole number) within the prime factors.
  3. Take the square root of the perfect squares and place them outside the radical sign.
  4. Multiply any remaining prime factors inside the radical sign.

3. What is the difference between a square root and a cube root?

A square root is the number that, when multiplied by itself, gives the original number. A cube root is the number that, when multiplied by itself three times, gives the original number. In other words, a square root is the inverse operation of squaring a number, while a cube root is the inverse operation of cubing a number.

4. How do you solve equations involving roots of real numbers?

To solve equations involving roots of real numbers, you can follow these steps:

  1. Isolate the radical expression on one side of the equation.
  2. Square both sides of the equation to eliminate the radical.
  3. Solve for the variable.
  4. Check your solution by plugging it back into the original equation.

5. Can you take the square root or cube root of a negative number?

Yes, you can take the square root or cube root of a negative number. However, the result will be a complex number. For example, the square root of -4 is 2i (2 times the imaginary unit, √-1). Similarly, the cube root of -27 is 3i (3 times the imaginary unit). This is because when you square or cube a negative number, the result is a positive number, and taking the square root or cube root of that positive number gives you both the positive and negative solutions.

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