Sure! Let's simplify the expression [tex]\( x^{b-c} \times x^{c-a} \times x^{a-b} \)[/tex].
Step-by-Step Solution:
1. Understand the Exponents Property:
Recall the property of exponents: when you multiply terms with the same base, you can add the exponents. In general, [tex]\( x^m \times x^n = x^{m+n} \)[/tex].
2. Combine the Exponents:
Here, the base [tex]\( x \)[/tex] is the same for all the terms, so we can combine the exponents:
[tex]\[
x^{b-c} \times x^{c-a} \times x^{a-b} = x^{(b-c) + (c-a) + (a-b)}
\][/tex]
3. Simplify the Exponents:
Now, add the exponents together:
[tex]\[
(b-c) + (c-a) + (a-b)
\][/tex]
4. Combine Like Terms:
When you add these exponents, observe that each of the terms [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( a \)[/tex] will cancel out:
[tex]\[
b - c + c - a + a - b
\][/tex]
5. Simplification:
Notice that [tex]\( -c + c \)[/tex] cancels out to 0, and [tex]\( -a + a \)[/tex] also cancels out to 0, and [tex]\( b - b \)[/tex] cancels out to 0:
[tex]\[
b - b + (-c + c) + (-a + a) = 0
\][/tex]
6. Conclusion:
Since the exponents sum up to 0, the simplified exponent becomes:
[tex]\[
x^0
\][/tex]
7. Final Answer:
Recall that any number (except zero) raised to the power of 0 is 1:
[tex]\[
x^0 = 1
\][/tex]
So, the simplified expression is:
[tex]\[
x^{b-c} \times x^{c-a} \times x^{a-b} = 1
\][/tex]
Thus, the simplified form of [tex]\( x^{b-c} \times x^{c-a} \times x^{a-b} \)[/tex] is [tex]\( 1 \)[/tex].
