Sure! Let's go through each of the expressions step by step to simplify them.
### Expression a:
[tex]\[ x^{b-c} \times x^{c-a} \times x^{a-b} \][/tex]
To simplify this, we can use the properties of exponents, specifically [tex]\( x^m \times x^n = x^{m+n} \)[/tex].
So, we add the exponents:
[tex]\[ (b-c) + (c-a) + (a-b) \][/tex]
Combine like terms:
[tex]\[ b - c + c - a + a - b \][/tex]
You'll notice that each term cancels out:
[tex]\[ 0 \][/tex]
Thus, the expression simplifies to:
[tex]\[ x^0 \][/tex]
And since [tex]\( x^0 = 1 \)[/tex] for any [tex]\( x \neq 0 \)[/tex]:
[tex]\[ 1 \][/tex]
### Expression b:
[tex]\[ \left(x^a\right)^{b-c} \times\left(x^b\right)^{c-a} \times\left(x^c\right)^{a-b} \][/tex]
We can use the property [tex]\( (x^m)^n = x^{mn} \)[/tex]:
[tex]\[ x^{a(b-c)} \times x^{b(c-a)} \times x^{c(a-b)} \][/tex]
Again, we add the exponents:
[tex]\[ a(b-c) + b(c-a) + c(a-b) \][/tex]
By distributing:
[tex]\[ ab - ac + bc - ab + ac - bc \][/tex]
You'll notice that these terms also cancel out:
[tex]\[ 0 \][/tex]
So, the expression simplifies to:
[tex]\[ x^0 \][/tex]
And:
[tex]\[ x^0 = 1 \][/tex]
Thus, the expression simplifies to:
[tex]\[ 1 \][/tex]
### Expression c:
[tex]\[ \left(x^{a-b}\right)^c \times\left(x^{b-c}\right)^a \times\left(x^{c-a}\right)^b \][/tex]
We can use the property [tex]\( (x^m)^n = x^{mn} \)[/tex]:
[tex]\[ x^{c(a-b)} \times x^{a(b-c)} \times x^{b(c-a)} \][/tex]
Notice that this is the same set of terms as in Expression b, but rearranged. Therefore, it simplifies similarly by adding the exponents:
[tex]\[ c(a-b) + a(b-c) + b(c-a) \][/tex]
This whole addition similarly simplifies to 0:
[tex]\[ 0 \][/tex]
So the expression simplifies to:
[tex]\[ x^0 \][/tex]
And:
[tex]\[ x^0 = 1 \][/tex]
Thus, the expression simplifies to:
[tex]\[ 1 \][/tex]
### Expression d:
[tex]\[ \left(x^{a+b}\right)^{a-b} \times\left(x^{b+c}\right)^{b-c} \times\left(x^{c+a}\right)^{c-a} \][/tex]
We can use the property [tex]\( (x^m)^n = x^{mn} \)[/tex]:
[tex]\[ x^{(a+b)(a-b)} \times x^{(b+c)(b-c)} \times x^{(c+a)(c-a)} \][/tex]
Now we expand each product in the exponents:
[tex]\[ (a + b)(a - b) = a^2 - b^2 \][/tex]
[tex]\[ (b + c)(b - c) = b^2 - c^2 \][/tex]
[tex]\[ (c + a)(c - a) = c^2 - a^2 \][/tex]
So the expression becomes:
[tex]\[ x^{a^2 - b^2} \times x^{b^2 - c^2} \times x^{c^2 - a^2} \][/tex]
Now, we add the exponents:
[tex]\[ (a^2 - b^2) + (b^2 - c^2) + (c^2 - a^2) \][/tex]
You'll notice that, once again, all terms cancel out:
[tex]\[ 0 \][/tex]
So the expression simplifies to:
[tex]\[ x^0 \][/tex]
And:
[tex]\[ x^0 = 1 \][/tex]
Thus, the expression simplifies to:
[tex]\[ 1 \][/tex]
To summarize:
a) [tex]\( 1 \)[/tex]
b) [tex]\( 1 \)[/tex]
c) [tex]\( 1 \)[/tex]
d) [tex]\( 1 \)[/tex]
