Simplify each expression. x 2 − 16 16 − x 2 \frac{x^2-16}{16-x^2}

So here we're asked to simplify the expression. Notice that we have x squared minus 16 over 16 minus x squared. Now it may be tempting to just try and break this whole thing down and factor it, but it actually turns out there's something much more simple that we can do. What I can do is change the order of either the numerator or denominator, and I'm gonna do this for the denominator. So keep the numerator the same, x squared minus 16, and then in the denominator I'm gonna put this negative x squared out front, and then we're going to have plus then we have the positive 16. Now the reason I'm writing it like this is notice we have opposite factors, we have a positive x squared and a negative 16, and then a negative x squared and a positive 16. So we know when we have opposite factors, the rule of thumb is this all just reduces down to negative one. So negative one would be the answer. As you can see, when you know how to deal with opposite factors, you don't have to go through this intensive factoring process in the numerator and denominator. You can just recognize the opposite factors and what it reduces to. Hope you found this helpful.

Simplify each expression.x2−1616−x2\frac{$$x^2-16$$}{$$16-x^2$$}

16x2−16\frac{16}{$$x^2-16$$}x2−1616

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1. Review of Real Numbers2h 39m

Simplifying Fractions
30m

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15m

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16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
40m

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Solving Linear Equations
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Simplifying Exponential Expressions Using All Exponent Rules
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8. Rational Expressions and Equations

Simplifying Rational Expressions

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Simplifying Rational Expressions

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Multiple Choice

Simplify each expression.
$\frac{x^{2} - 16}{16 - x^{2}}$

$- x^{2}$

$-1$

$x^2$

$\frac{16}{x^2-16}$

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Verified step by step guidance

Start with the given expression: $\frac{x^2 - 16}{16 - x^2}$.

Recognize that both the numerator and denominator are differences of squares. Recall the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$.

Factor the numerator: $x^2 - 16 = (x - 4)(x + 4)$.

Factor the denominator: \$16 - x^2 = (4 - x)(4 + x)$. Notice that $4 - x$ can be rewritten as $-(x - 4)$ to help with simplification.

Rewrite the denominator using this idea: \$16 - x^2 = -(x - 4)(x + 4)$. Now the expression is \(\frac{(x - 4)(x + 4)}{-(x - 4)(x + 4)}$. Cancel the common factors \)(x - 4)(x + 4)$, leaving $-1$ as the simplified result.

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Simplify each expression.x2−1616−x2\(\frac{x^2-16}{16-x^2}\)

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Simplify each expression.
$\frac{x^{2} - 16}{16 - x^{2}}$