Week 3: Indices & Quadratics

The Challenge

Solve for $x$ by simplifying the indices first:

$$ \sqrt[3]{5^{3x^2 - 3x}} = \sqrt[4]{25^{4x+20}} $$

Step 1: Strategic Alignment

Look at the bases: We have $5$ on the Left and $25$ on the Right. To compare the exponents, the bases must match.

Decision Time: Which base is easier to work with?

Step 2: Deal with the Radicals

Now that the bases match, we still have roots. Recall that $\sqrt[n]{x} = x^{\frac{1}{n}}$.

Question: How can you rewrite the radicals as fractional exponents?

LHS exponent simplifies to:

RHS exponent simplifies to:

Step 3: The Hidden Quadratic

If $5^A = 5^B$, then $A=B$.

Task: Set your exponents equal and rearrange them into the standard Quadratic form $Ax^2 + Bx + C = 0$.

$x^2 +$ $x +$ $= 0$

Step 4: Conclusion

Solve the quadratic to find the values for $x$.

$x_1 =$ $x_2 =$

🎉 Investigation Complete!

You realized that a complex radical equation was actually just a Quadratic in disguise.

Variation: Intersection of Curves

Mathematical Modeling: Instead of solving an equation, imagine finding where two functions meet.

$$ f(x) = \sqrt[3]{8^{x^2}} \quad \text{and} \quad g(x) = \frac{16}{2^{-2x}} $$

Visual Analysis: Use the graph below to estimate the intersection point ($x > 0$).

(Drag the graph to see where the lines cross! Look for x approx 3.2)

Step 1: Strategic Analysis

To intersect these functions, we set $f(x) = g(x)$. But the bases ($8$ and $2$) are different.

Question: What single base connects $8, 16,$ and $2$?

Common Base =

Step 2: Simplify Functions

Rewrite both functions as powers of your chosen base.

$f(x)$ exponent: (Hint: $8=2^3$)

$g(x)$ exponent:

Step 3: Solve the Intersection

Equate the exponents and solve. There are two intersection points. Find the one with a positive x-value.

$x =$

Excellent. Now you are ready to prove your skills.

🏠 Independent Practice

Apply the "Same Base Strategy" and "Quadratic Solving" to these three variations.

Variation 1: The Simple Radical

Solve for the positive value of $x$:

$$ \sqrt{3^{x^2}} = 9^{x-1} $$

Positive $x =$

Variation 2: Functional Notation

Given the functions $f(x)$ and $g(x)$ below, find the positive value of $x$ such that $f(x) = g(x)$.

$$ f(x) = \sqrt[3]{27^{x^2}}, \quad g(x) = 81 $$

Positive $x =$

Variation 3: Higher Roots & Indices (Challenge)

One side has a 4th root, the other is in index form. Find the largest value of $x$:

$$ \sqrt[4]{16^{x^2}} = 2^{4x-3} $$

Largest $x =$