Certainly! Let's go through each part step-by-step to solve them.
### Part b:
Given the equation:
[tex]\[ a^x = \left(a^p \times a^q\right) \div a^r \][/tex]
1. Convert the multiplication and division of exponents:
[tex]\[ a^x = \frac{a^p \cdot a^q}{a^r} \][/tex]
2. Apply the properties of exponents:
- [tex]\( a^p \cdot a^q = a^{p+q} \)[/tex]
- [tex]\( \frac{a^{p+q}}{a^r} = a^{p+q-r} \)[/tex]
So, the equation becomes:
[tex]\[ a^x = a^{p+q-r} \][/tex]
3. Since the bases are the same (and are non-zero), the exponents must be equal:
[tex]\[ x = p + q - r \][/tex]
Hence, [tex]\( x = p + q - r \)[/tex].
### Part c:
Given the equation:
[tex]\[ \left(b^m \div b^n\right) \times b^p = b^y \][/tex]
1. Simplify the expression inside the parentheses:
[tex]\[ b^m \div b^n = b^{m-n} \][/tex]
2. Multiply the result by [tex]\( b^p \)[/tex]:
[tex]\[ b^{m-n} \times b^p = b^{(m-n)+p} \][/tex]
3. Since the left-hand side equals [tex]\( b^y \)[/tex], and the bases are the same, we equate the exponents:
[tex]\[ y = (m-n) + p \][/tex]
Hence, [tex]\( y = m - n + p \)[/tex].
### Part d:
Given the equation:
[tex]\[ (x^m)^n = x^m \cdot x_{X^n} \][/tex]
Let's carefully examine the interpretation. We suspect there may be a typographical error here especially with the variable notation [tex]\( x_{X^n} \)[/tex]. For the sake of clarity, let's assume a probable correct form of the equation:
[tex]\[ (x^m)^n = x^{mn} \][/tex]
If this is true, then the equation simplifies as:
[tex]\[ x^{mn} = x^m \cdot x^{X^n} \][/tex]
4. Using the properties of exponents, particularly [tex]\( x^a \cdot x^b = x^{a+b} \)[/tex], we get:
[tex]\[ x^{mn} = x^{m + X^n} \][/tex]
5. As the bases are the same, equate the exponents:
[tex]\[ mn = m + X^n \][/tex]
Attempt to isolate [tex]\( m \)[/tex] in terms of [tex]\( n \)[/tex]:
[tex]\[ mn - m = X^n \][/tex]
6. Factor out [tex]\( m \)[/tex] from the left-hand side:
[tex]\[ m(n-1) = X^n \][/tex]
Thus, expressing [tex]\( m \)[/tex] in terms of [tex]\( n \)[/tex]:
[tex]\[ m = \frac{X^n}{n-1} \][/tex]
