1)find rational number a and b such that 2+5root7/2-5root7=a+root7 b
2)find the value of x if 5 pwer 2 square 9=(125)x
3)if a=9 b=root17 then find the value of (a power2- b power2)-1/3
You have to use brackets
Surely you meant:
(2+5√7)/(2-5√7) = a + √7 b
on the left side, multiply by (2+5√7)/(2+5√7)
then match up the coefficients of the rational terms and match the coefficients of the irrational terms
let me know what you got
2. I don't know what "5 pwer 2 square 9" means.
3.
(a^2 - b^2) - 1/2
= (81 - 17) - 1/3
= .... continue
5^(2√9) = 125^x
5^(2*3) = 125^x
(5^3)^2 = 125^x
125^2 = 125^x
2 = x
1) To find rational numbers a and b such that 2 + (5√7) / (2 - 5√7) = a + √7b, we need to rationalize the denominator.
Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2 - 5√7 is 2 + 5√7.
(2 + 5√7) * (2 + 5√7) = 4 + 20√7 + 25 * 7 = 189 + 20√7
(2 - 5√7) * (2 + 5√7) = 4 - 25 * 7 = -171
Step 2: Simplify the expression.
(2 + (5√7)) / (2 - 5√7) = (2 + (5√7)) * (2 + 5√7) / (2 - 5√7) * (2 + 5√7)
= (189 + 20√7) / -171
= -189/171 - 20√7/171
= -7/6 - (20/171)√7
Therefore, a = -7/6 and b = -20/171.
2) To find the value of x in the equation 5^2^(9) = (125)x, we need to simplify the expression using the rules of exponents.
Step 1: Simplify the exponent.
2^9 = (2^3)^3 = 8^3 = 512
Step 2: Rewrite the equation with the simplified exponent.
5^512 = (125)x
Step 3: Convert both bases to the same value.
5 = 5^1
125 = 5^3
Step 4: Equate the exponents.
5^1 * 5^3 = 5^512
Step 5: Use the property of exponents that states a^m * a^n = a^(m+n).
5^(1+3) = 5^512
Step 6: Simplify the exponent.
5^4 = 5^512
Since the bases are the same, the exponents must be equal. Therefore,
4 = 512
This is not true, so there is no real solution for x in this equation.
3) To find the value of (a^2 - b^2)^(1/3) where a = 9 and b = √17, we substitute the given values into the expression.
(a^2 - b^2)^(1/3) = (9^2 - (√17)^2)^(1/3)
= (81 - 17)^(1/3)
= 64^(1/3)
Step 1: Compute the cube root of 64.
64^(1/3) = 4
Therefore, the value of (a^2 - b^2)^(1/3) is 4.