Simplifying Algebraic Expressions with Exponents
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6. Let's simplify.
a) $x^{a-b} \times x^{b-a}$
b) $x^{a-b} \times x^{b-c} \times x^{c-a}$
c) $(x^a)^{b-c} \times (x^b)^{c-a} \times (x^c)^{a-b}$
d) $(x^{p-q})^r \times (x^{q-r})^p \times (x^{r-p})^q$
e) $\frac{x^{a+b} \times x^{b+c} \times x^{c+a}}{x^{2a} \times x^{2b} \times x^{2c}}$
f) $\frac{(x^2)^{a+b} \times (x^2)^{b+c} \times (x^2)^{c+a}}{(x^a \times x^b \times x^c)^4}$
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Step by Step Written Solution
Hi Dharma, let's solve this together. We will simplify two interesting exponent problems from your worksheet: parts c and f, using the fundamental laws of indices.
Laws of Indices
We will use these core exponent rules:
1. Power of a Power: $(x^m)^n = x^{m \cdot n}$
2. Product Rule: $x^m \times x^n = x^{m+n}$
Let's start with part c. The expression consists of multiple bases of x, each raised to a power and then raised to another variable exponent.
Simplifying Part c)
First, we apply the power of a power rule to each of the three factors. We multiply the inner exponent by the outer exponent.
Now, let's distribute the multiplication in each exponent. This gives us x to the power of a b minus a c, times x to the power of b c minus a b, times x to the power of a c minus b c.
Since the bases are all x, we can multiply these terms together by adding all of their exponents.
Let's simplify the sum in the exponent. Notice that positive a b and negative a b cancel out, negative a c and positive a c cancel out, and positive b c and negative b c cancel out, leaving zero.
Since any non-zero base raised to the power of zero is equal to one, the expression simplifies to one.
Now let's tackle the more complex-looking part f. This expression has exponents in both the numerator and the denominator.
Simplifying Part f)
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