Alright, let's simplify the given expressions step-by-step.
### Part a)
We need to simplify the expression:
[tex]\[ x^{b-c} \times x^{c-a} \times x^{a-b} \][/tex]
1. Combine the exponents using the property of exponents: [tex]\( x^m \times x^n = x^{m+n} \)[/tex]:
[tex]\[ x^{(b-c)} \times x^{(c-a)} \times x^{(a-b)} = x^{(b-c + c-a + a-b)} \][/tex]
2. Simplify the exponent:
[tex]\[ b - c + c - a + a - b \][/tex]
3. Notice that every term cancels out:
[tex]\[ b - b + c - c + a - a = 0 \][/tex]
4. So the expression becomes:
[tex]\[ x^0 \][/tex]
5. Any number to the power of 0 is 1:
[tex]\[ x^0 = 1 \][/tex]
Hence, the simplified form of [tex]\( x^{b-c} \times x^{c-a} \times x^{a-b} \)[/tex] is [tex]\( \boxed{1} \)[/tex].
### Part c)
We need to simplify the expression:
[tex]\[ \left(x^{a-b}\right)^c \times \left(x^{b-c}\right)^a \times \left(x^{c-a}\right)^b \][/tex]
1. Apply the power of a power property: [tex]\( (x^m)^n = x^{mn} \)[/tex]:
[tex]\[ \left(x^{a-b}\right)^c = x^{(a-b)c} \][/tex]
[tex]\[ \left(x^{b-c}\right)^a = x^{(b-c)a} \][/tex]
[tex]\[ \left(x^{c-a}\right)^b = x^{(c-a)b} \][/tex]
2. Combine the exponents using the property of exponents: [tex]\( x^m \times x^n = x^{m+n} \)[/tex]:
[tex]\[ x^{(a-b)c} \times x^{(b-c)a} \times x^{(c-a)b} = x^{(a-b)c + (b-c)a + (c-a)b} \][/tex]
3. Simplify the exponent:
[tex]\[ (a-b)c + (b-c)a + (c-a)b \][/tex]
4. Distribute and combine like terms:
[tex]\[ ac - bc + ab - ac + bc - ab \][/tex]
5. Notice that every term cancels out:
[tex]\[ ac - ac + bc - bc + ab - ab = 0 \][/tex]
6. So the expression becomes:
[tex]\[ x^0 \][/tex]
7. Any number to the power of 0 is 1:
[tex]\[ x^0 = 1 \][/tex]
Hence, the simplified form of [tex]\( \left(x^{a-b}\right)^c \times \left(x^{b-c}\right)^a \times \left(x^{c-a}\right)^b \)[/tex] is [tex]\( \boxed{1} \)[/tex].
So the results are:
[tex]\[ \boxed{1} \text{ for both expressions.} \][/tex]
