- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Power functions and first sketches practice question 13
Find the value of at which the curves below intersect.
Both sides are powers of , one in disguise. How can you gather everything into a single power of equal to a number?
Write as , then divide both sides by (allowed, since ).
You should reach . Raise both sides to the power .
An exact expression, or a decimal to three decimal places.
Theory UBC Math 100 §1.1Key results
Worked solution
Strategy
Both curves are power functions, so set them equal and use the laws of exponents to isolate a single power of x. Because x is positive, dividing by a power of x is safe.
Working
Write the square root as a fractional power so the exponent laws apply.
Divide by x1/2. This would lose x = 0, but the question asks for x > 0.
Raise both sides to the reciprocal power 2/3.
Both sides agree, so the curves really meet there.
Answer
The answer is 4 to the power two-thirds, which is the cube root of 16, about 2.520.
Check
At x = 2.52, 4√x ≈ 6.35 and x² ≈ 6.35, so the two curves do meet there.
Where this goes wrong
Undoing the wrong exponent at the end. Squaring both sides is fine and gives 16x = x⁴, so x³ = 16, but the answer is then written as 161/4 or ∛4. The exponent left on x is 3, so x = ∛16.
Try a similar question
Find the positive value of at which the curves below intersect.Saved in this browser only.