Find x, given 5(√5)(x+5) = (√5)(2x+7)

Concept

When an equation has powers on both sides, rewrite every term with the SAME base; once the bases are identical, the exponents must be equal. A root is a fractional power: √5 = 51/2, and a leading numerical factor like 5 is 51, which folds into the exponent of that base.

Application

1. Write each side as a power of 5. Left side: 5·(√5)(x+5) = 51·5(x+5)/2 = 51 + (x+5)/2.

2. Right side: (√5)(2x+7) = 5(2x+7)/2.

3. Equal bases ⇒ equal exponents: 1 + (x+5)/2 = (2x+7)/2.

4. Multiply through by 2: 2 + (x+5) = 2x+7, i.e. x + 7 = 2x + 7.

5. Subtract x+7 from both sides: 0 = x, so x = 0.

Cross-check

Substitute x = 0: left = 5·(√5)5 = 5·55/2 = 57/2; right = (√5)7 = 57/2. Both sides equal 57/2, confirming x = 0.