Explain the process of raising each side of an equation to the nth power.
Understand the Problem
The question asks for an explanation of the process of raising both sides of an equation to the nth power. This involves understanding how exponents work and how they apply to maintaining equality in an equation.
Answer
If $a = b$, then $a^n = b^n$.
Answer for screen readers
Raising both sides of an equation $a=b$ to the $n$th power results in the new equation $a^n = b^n$. This operation maintains the equality as long as the same power is applied to both sides.
Steps to Solve
- Understanding the nth power
Raising a number to the nth power (where n is an integer) means multiplying that number by itself n times. For example, $x^3 = x \cdot x \cdot x$.
- Raising both sides of an equation to the nth power
If we have an equation $a = b$, raising both sides to the nth power means we are performing the same operation on both sides, which maintains the equality. This results in the new equation $a^n = b^n$.
- Illustrative example
Let's say we have the equation $x = 2$. If we raise both sides to the power of 3 (i.e., $n = 3$), we get $x^3 = 2^3$, which simplifies to $x^3 = 8$. The equality is maintained.
- Generalization
In general, if $a = b$, then $a^n = b^n$ for any integer $n$. This is a fundamental property used in solving many algebraic equations, especially when dealing with radicals.
Raising both sides of an equation $a=b$ to the $n$th power results in the new equation $a^n = b^n$. This operation maintains the equality as long as the same power is applied to both sides.
More Information
Raising both sides of an equation to the $n$th power is a common technique used to eliminate radicals (like square roots, cube roots, etc.) from equations. For example, if you have $\sqrt{x} = 3$, squaring both sides (raising to the power of 2) gives you $(\sqrt{x})^2 = 3^2$, which simplifies to $x = 9$. But be cautious of introducing extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation.
Tips
A common mistake is forgetting to apply the exponent to all terms on each side of the equation, especially if one or both sides involve sums or differences. For example, if you have $x + 1 = 3$, squaring both sides should result in $(x + 1)^2 = 3^2$, which expands to $x^2 + 2x + 1 = 9$, not $x^2 + 1 = 9$.
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