Problem 75 Solve each equation. $$(t-1)^{... [FREE SOLUTION] (2024)

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Understanding exponent rules is crucial when solving algebraic equations. These rules help us manipulate and simplify expressions with exponents. Here are some essential exponent rules:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \(\text{(a}^m\text{)}^n = a^{mn}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Product: \(\text{(ab)}^n=a^n\cdot b^n\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)}

In our exercise, we used the power of a power rule to simplify \(\text{((t-1)}^{-2/3})^{-3/2}\right)\), which equals \(\text{t-1}\). Learning these rules makes working with exponents much easier!

Fractional exponents might seem tricky at first, but they're simply another way to represent roots and powers. A fractional exponent \(a^{m/n}\) means the \(n\)th root of \(a\) raised to the \(m\)th power. For example, \(8^{2/3}\) means \(\sqrt[3]{8^2}\). Here's a quick breakdown:

• The denominator (bottom number) of the fraction tells you the root.
• The numerator (top number) tells you the power to which the base is raised.
  For our exercise, we simplified \(\text{(t-1)}^{-2/3}\) by raising both sides to the power of \-3/2\. This \cancelled\ the fractions, simplifying the equation steps later.

Remember, practice makes perfect! The more you work with these, the easier they'll become!