Question

If \[{a^x} = bc\], \[{b^y} = ac\], and \[{c^z} = ab\]. Then calculate the value of \[xyz\].
A. 0
B. 1
C. \[x + y + z + 2\]
D. \[x + y + z\]

Answer 1

Hint: First we will raise to power \[y\] both sides of \[{a^x} = bc\] and simplify it. Then substitute \[{b^y} = ac\]. Again, we will raise to power \[z\]. Then put the value \[{c^z}\]. Then we will use the indices \[{a^m} \cdot {a^n} = {a^{m + n}}\] to simplify it and plug \[bc = {a^x}\]. At least we compare the power of \[a\]of the equation to get the value of \[xyz\].

Formula Used:
\[{a^m} \cdot {a^n} = {a^{m + n}}\]
\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
\[{\left( {ab} \right)^m} = {a^m}{b^m}\]

Complete step by step solution:
Given that,
\[{a^x} = bc\], \[{b^y} = ac\], and \[{c^z} = ab\]
Now raise to power \[y\] on both sides of \[{a^x} = bc\]
\[{\left( {{a^x}} \right)^y} = {\left( {bc} \right)^y}\]
Applying the formulas \[{\left( {{a^m}} \right)^n} = {a^{mn}}\] and \[{\left( {ab} \right)^m} = {a^m}{b^m}\]
\[ \Rightarrow {a^{xy}} = {b^y}{c^y}\]
Putting \[{b^y} = ac\]
\[ \Rightarrow {a^{xy}} = ac \cdot {c^y}\]
Now raise to power \[z\] on both sides of the equation
\[ \Rightarrow {\left( {{a^{xy}}} \right)^z} = {\left( {ac \cdot {c^y}} \right)^z}\]
Applying the formula \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
\[ \Rightarrow {\left( {{a^{xy}}} \right)^z} = {a^z}{c^z}{c^{yz}}\]
\[ \Rightarrow {\left( {{a^{xy}}} \right)^z} = {a^z}{c^z}{\left( {{c^z}} \right)^y}\]
Now putting \[{c^z} = ab\]
\[ \Rightarrow {a^{xyz}} = {a^z} \cdot \left( {ab} \right) \cdot {\left( {ab} \right)^y}\]
Applying the formula \[{\left( {ab} \right)^m} = {a^m}{b^m}\]
\[ \Rightarrow {a^{xyz}} = {a^z} \cdot a \cdot b \cdot {a^y} \cdot {b^y}\]
Apply the formula \[{a^m} \cdot {a^n} = {a^{m + n}}\]
\[ \Rightarrow {a^{xyz}} = {a^{z + y + 1}} \cdot {b^y} \cdot b\]
Now putting \[{b^y} = ac\]
\[ \Rightarrow {a^{xyz}} = {a^{z + y + 1}} \cdot ac \cdot b\]
Apply the formula \[{a^m} \cdot {a^n} = {a^{m + n}}\]
\[ \Rightarrow {a^{xyz}} = {a^{z + y + 2}} \cdot bc\]
Now putting \[bc = {a^x}\]
\[ \Rightarrow {a^{xyz}} = {a^{z + y + 2}} \cdot {a^x}\]
Apply the formula \[{a^m} \cdot {a^n} = {a^{m + n}}\]
\[ \Rightarrow {a^{xyz}} = {a^{x + z + y + 2}}\]
Now compare the power \[a\]
\[ \Rightarrow xyz = x + z + y + 2\]

Hence option B is the correct option.

Note:Students often confused with the formulas \[{\left( {{a^m}} \right)^n} = {a^{mn}}\] and \[{\left( {{a^m}} \right)^n} = {a^{m + n}}\]. The correct formulas are \[{\left( {{a^m}} \right)^n} = {a^{mn}}\] and \[{a^m} \cdot {a^n} = {a^{m + n}}\].